Functoriality of the Schmidt construction

After proving, in a purely categorial way, that the inclusion functor InAlg(Σ) from Alg(Σ), the category of many-sorted Σ-algebras, to PAlg(Σ), the category of many-sorted partial Σ-algebras, has a left adjoint FΣ, the (absolutely) free completion functor, we recall, in connection with the functor FΣ, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next we define a category Cmpl(Σ), of Σ-completions, and prove that FΣ, labeled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this we associate to an ordered pair (α, f ), where α = (K, γ, α) is a morphism of Σ-completions from F = (C, F, η) to G = (D, G, ρ) and f a homomorphism in D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism ΥG,0 α ( f ) : Schα( f ) // B. We then prove that there exists an endofunctor, ΥG,0 α , of Mortw(D), the twisted morphism category of D, thus showing the naturalness of the previous construction. Afterwards we prove that, for every Σ-completion G = (D, G, ρ), there exists a functor Υ from the comma category (Cmpl(Σ)↓G) to End(Mortw(D)), the category of endofunctors of Mortw(D), such that ΥG,0, the object mapping of Υ, sends a morphism of Σ-completion in Cmpl(Σ) with codomain G, to the endofunctor ΥG,0 α .


Introduction
Both partiality and heterogeneity are phenomena that occur, since time immemorial, in mathematics and in computability theory. The study of both topics, from an abstract point of view, is carried out, mainly but not exclusively, by means of the many-sorted partial algebras and the homomorphisms between them, i.e., its study takes place in the category PAlg(Σ), of many-sorted partial Σ-algebras, and subcategories of it. However, such a study is not limited to such categories, but also includes the investigation of the functors from and to such categories, as well as the natural transformations between them. In this regard, there is a key result: The inclusion functor In Alg(Σ) from Alg(Σ), the category of many-sorted Σ-algebras, to PAlg(Σ) has a left adjoint F Σ , the (absolutely) free completion functor (see [3], [4] and [17]) (and, for every partial Σ-algebra A, η A , the value of the unit of the adjunction at A, is a dense injective homomorphism from A into F Σ (A)). And, in connection with F Σ , we have the generalized recursion theorem of Schmidt (see [17]), which we will also call the Schmidt construction, and which states the following: In Sections 2 to 5 we fix notation and terminology and review those concepts and results about many-sorted sets, many-sorted algebras, many-sorted partial algebras and partial Dedekind-Peano algebras, which will be used in this paper and which will make it as self-contained as possible (this material is quite standard, so the expert reader may skip most of it).
In Section 6, taking into account a theorem about adjoint functors, we again obtain, as a corollary of it, the theorem that states that the inclusion functor In Alg(Σ) from Alg(Σ) to PAlg(Σ) has a left adjoint F Σ , the (absolutely) free completion functor. At the end of this section, and related to the functor F Σ from PAlg(Σ) to Alg(Σ), we state the generalized recursion theorem of Schmidt (see [17]), which we will also call the Schmidt construction. We remark that the rest of this article will focus on such a nice and fundamental mathematical construction in order to show its functoriality.
Next, in Section 7, and assuming that the category PAlg(Σ) is equipped with a factorization system (E, M), we define a category Cmpl(Σ), of Σ-completions, whose objects are ordered triples F = (C, F, η) that satisfy the following conditions: 1.
C is a wide subcategory of PAlg(Σ), i.e., a subcategory of PAlg(Σ) such that Ob(C) = Ob(PAlg(Σ)), satisfying that 1.1 for every homomorphism f of C if f = h • g, for some (g, h) ∈ E × M, then h is a homomorphism of C.

2.
F is a functor from C to Alg(Σ).

3.
η is a natural transformation from In C to In Alg(Σ) • F, i.e., for every homomorphism f : A / / B between partial Σ-algebras of C, we have that Moreover, we prove that F Σ = (PAlg(Σ), F Σ , η), the Σ-completion associated to the free completion functor F Σ and the unit η of the adjunction F Σ In Alg(Σ) , is a weakly initial object of Cmpl(Σ).
Following this, in Section 8, we begin to realize our project of showing the functoriality of the Schmidt construction. To do it, we begin by associating to an ordered pair (α, f ), where α = (K, γ, α) is a morphism of Σ-completions from F = (C, F, η) to G = (D, G, ρ) and f a homomorphism of D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism Υ G,0 α ( f ) : Sch α ( f ) / / B. This construction, actually, generalizes the Schmidt construction. We then prove that there exists an endofunctor, denoted Υ G,0 α , called the Schmidt endofunctor relative to G and α, of Mor tw (D), the twisted morphism category of D, thus showing the naturalness of the Schmidt construction.
Afterwards, in Section 9, we prove that, for every Σ-completion G, there exists a functor Υ G from the comma category (Cmpl(Σ)↓G) to End(Mor tw (D)), the category of endofunctors of Mor tw (D), such that Υ G,0 , the object mapping of Υ G , sends an object α : F / / G of (Cmpl(Σ)↓G), i.e., a morphism of Σ-completion of Cmpl(Σ) with codomain G, to the endofunctor Υ G,0 α of End(Mor tw (D)). Although there are excellent monographs and survey articles on many-sorted partial algebras, e.g., [5] and [6], unfortunately it does not seem that, generally speaking, researchers are familiar enough with such a mathematical field. So it has seemed to us appropriate, for completeness and easy reference, to recall those fundamental notions and constructions in that field of which we make use.
Our underlying set theory is ZFSk, Zermelo-Fraenkel-Skolem set theory (also known as ZFC, i.e., Zermelo-Fraenkel set theory with the axiom of choice) plus the existence of a Grothendieck universe U, fixed once and for all (see [14], pp. 21-24). We recall that the elements of U are called U-small sets and the subsets of U are called U-large sets or classes. The image of f , denoted by Im( f ), is f [A], i.e., ( f s [A s ]) s∈S . Moreover, if X ⊆ A, then the restriction of f to X, denoted by f X , f | X or res X ( f ), is f • in X,A , where in X,A = (in X s ,A s ) s∈S is the canonical embedding of X into A.

Remark 1.
To give an S-sorted mapping from A to B, as in the just stated definition, is equivalent to give an S-indexed family f = ( f s ) s∈S , where, for every s in S, f s is a mapping from A s to B s . Thus, an S-sorted mapping from A to B is, essentially, an element of ∏ s∈S Hom(A s , B s ).

Definition 4.
Let I be a set in U and (A i ) i∈I an I-indexed family of S-sorted sets. Then the product of (A i ) i∈I , denoted by ∏ i∈I A i , is the S-sorted set defined, for every s ∈ S, as ∏ i∈I A i s = ∏ i∈I A i s , where ∏ i∈I A i s = (a i ) i∈I ∈ Fnc I, i∈I A i s | ∀ i ∈ I a i ∈ A i s .
For every i ∈ I, the i-th canonical projection, pr i = (pr i s ) s∈S , is the S-sorted mapping from ∏ i∈I A i to A i that, for every s ∈ S, sends (a i ) i∈I in ∏ i∈I A i s to a i in A i s . The ordered pair (∏ i∈I A i , (pr i ) i∈I ) has the following universal property: For every S-sorted set B and every Iindexed family of S-sorted mappings ( f i ) i∈I , where, for every i ∈ I, f i is an S-sorted mapping from B to A i , there exists a unique S-sorted mapping f i i∈I from B to ∏ i∈I A i such that, for every i ∈ I, pr i • f i i∈I = f i . The coproduct of (A i ) i∈I , denoted by i∈I A i , is the S-sorted set defined, for every s ∈ S, as i∈I A i s = i∈I A i s , where i∈I A i s = i∈I (A i s × {i}).
For every i ∈ I, the i-th canonical injection, in i , is the S-sorted mapping from A i to i∈I A i that, for every s ∈ S, sends a in A i s to (a, i) in i∈I A i s . The ordered pair ( i∈I A i , (in i ) i∈I ) has the following universal property: For every S-sorted set B and every I-indexed family of S-sorted mappings ( f i ) i∈I , where, for every i ∈ I, f i is an S-sorted mapping from A i to B, there exists a unique S-sorted mapping [ f i ] i∈I from i∈I A i to B such that, for every i ∈ I, The remaining set-theoretic operations on S-sorted sets: (binary coproduct), (union), ∪ (binary union), (intersection), ∩ (binary intersection), − (difference) and A (complement of an S-sorted set in a fixed S-sorted A), are defined in a similar way, i.e., componentwise. Definition 5. We will denote by 1 S the (standard) final S-sorted set of Set S , which is 1 S = (1) s∈S , and by ∅ S the initial S-sorted set, which is ∅ S = (∅) s∈S . We shall abbreviate 1 S to 1 and ∅ S to ∅ when this is unlikely to cause confusion. Proposition 1. Let B be an S-sorted set, Y a subset of B and f an S-sorted mapping from A to B. Then the following statements are equivalent: 1.
There exists an S-sorted mapping h from A to Y such that the diagram in Figure 1 commutes.
If one of the above equivalent statements holds, then we will call h, which is univocally determined, the corestriction of f to Y and we denote it by f | Y or cores Y ( f ). Proposition 2. Let PSet S be the category whose objects are the S-sorted set pairs, i.e., the ordered pairs pairs (A, X) where A is an S-sorted set and X ⊆ A, and in which the set of morphisms from (A, X) to (B, Y) is the set of all S-sorted mappings f from A to B such that f [X] ⊆ Y. Let G be the functor from Set S to PSet S whose object mapping sends A to (A, A) and whose morphism mapping sends f : Then we denote by f | Y X , bires X,Y ( f ), or, if no confusion can arise, f the S-sorted mapping cores Y (res X ( f )) (which is identical to res X (cores Y ( f ))). We will call this S-sorted mapping the birestriction of f to X and Y.
Definition 7. Let A be an S-sorted set. Then the cardinal of A, denoted by card(A), is card( A), i.e., the cardinal of the set A = s∈S (A s × {s}). An S-sorted set A is finite if card(A) < ℵ 0 . We will say that an S-sorted set X is a finite subset of A if X is finite and X ⊆ A. We will denote by Sub f (A) the set of all S-sorted sets X in Sub(A) which are finite. Definition 8. Let A be an S-sorted set. Then the support of A, denoted by supp S (A), is the set { s ∈ S | A s = ∅ }.
Remark 2. An S-sorted set A is finite if and only if supp S (A) is finite and, for every s ∈ supp S (A), A s is finite.
We next define the notion of equivalence relation on a many-sorted set and state the universal property of the corresponding quotient many-sorted set. Definition 9. An S-sorted equivalence relation on (or, to abbreviate, an S-sorted equivalence on) an S-sorted set A is an S-sorted relation Φ on A such that, for every s ∈ S, Φ s is an equivalence relation on A s . We will denote by Eqv(A) the set of all S-sorted equivalences on A (which is an algebraic closure system on A × A), by Eqv(A) the algebraic lattice (Eqv(A), ⊆), by ∇ A the greatest element of Eqv(A) and by ∆ A the least element of Eqv(A). As for ordinary sets, Eqv(A) is also an algebraic lattice, and we denote by Eg A the canonically associated algebraic closure operator. For A, we have that For an S-sorted equivalence relation Φ on A, the S-sorted quotient set of A by where, for every s ∈ S and every x ∈ A s , [x] Φ s , the equivalence class of x with respect to Φ s (or, the Φ-equivalence class of x) is {y ∈ A s | (x, y) ∈ Φ s }, and pr Φ : A / / A/Φ, the canonical projection from A to A/Φ, is the S-sorted mapping (pr Φ s ) s∈S , where, for every s ∈ S, pr Φ s is the canonical projection from A s to A s /Φ s (which sends x in A s to pr Φ s (x) = [x] Φ s , the Φ s -equivalence class of x, in A s /Φ s ). Moreover, if Ψ is an S-sorted equivalence on an S-sorted set B and f an S-sorted mapping from A to B, then the astriction of f to B/Ψ, denoted by ast B/Ψ ( f ), is pr Ψ • f , where pr Φ is the canonical projection of B onto B/Ψ. Proposition 3. Let A be an S-sorted set, Φ an S-sorted equivalence on A and f : A / / B an S-sorted mapping. Then the following statements are equivalent: There exists an S-sorted mapping h from A/Φ to B such that the diagram in Figure 2 commutes. A where the involved S-sorted mappings are defined coordinatewise, i.e., for every s ∈ S, (pr Ker( f ) ) s is pr Ker( f s ) , the canonical projection from We next define the concept of free monoid on a set and several notions associated with it that will be used afterwards to construct the free algebra on an S-sorted set.
of words (w, v) on A to the mapping w v from |w| + |v| to A, where |w| and |v| are the lengths (≡ domains) of the mappings w and v, respectively, defined as follows: and λ, the empty word on A, is the unique mapping from ∅ to A. A word w ∈ A is usually denoted as a sequence (a i ) i∈|w| , where, for i ∈ |w|, a i is the letter in A satisfying w(i) = a i . We will denote by η A the mapping from A to A that sends a ∈ A to (a) ∈ A , i.e., to the mapping (a) : 1 / / A that sends 0 to a. The ordered pair (A , η A ) is a universal morphism from A to the forgetful functor from the category Mon, of monoids, to Set.

Many-sorted algebras
Our next aim is to provide those notions from the field of many-sorted universal algebra that will be used afterwards. We will specially focus on the constructive description of the subalgebra generating many-sorted operator, the congruence generating many-sorted operator and the (absolutely) free many-sorted algebra on a many-sorted set.

Convention.
In what follows, for a set of sorts S, an arbitrary word on S will be denoted by s, i.e., a lower case bold type s. The letter s will be used to represent an arbitrary letter in S. Definition 13. An S-sorted signature is a mapping Σ from S × S to U which sends a pair (s, s) ∈ S × S to the set Σ s,s of the formal operations of arity s, sort (or coarity) s and rank (or biarity) (s, s).
Assumption 2. From now on Σ stands for an S-sorted signature, fixed once and for all.
We shall now give precise definitions of the concepts of many-sorted algebra and homomorphism between many-sorted algebras.

Definition 14.
The S × S-sorted set of the finitary operations on an S-sorted set A is (Hom(A s , A s )) (s,s) where, for every s ∈ S , A s = ∏ j∈|s| A s j , with |s| denoting the length of the word s (if s = λ, then A λ is a final set). A structure of Σ-algebra on an S-sorted set A is a family (F s,s ) (s,s)∈S ×S , denoted by F, where, for (s, s) ∈ S × S, F s,s is a mapping from Σ s,s to Hom(A s , A s ) (if (s, s) = (λ, s) and σ ∈ Σ λ,s , then F λ,s (σ) picks out an element of A s ). For a pair (s, s) ∈ S × S and a formal operation σ ∈ Σ s,s , in order to simplify the notation, the operation F s,s (σ) from A s to A s will be written as F σ . A many-sorted Σ-algebra (or, to abbreviate, Σ-algebra) is a pair (A, F), denoted by A, where A is an S-sorted set and F a structure of Σ-algebra on A. A Σ-homomorphism (or, to abbreviate, homomorphism) from A to B, where B = (B, G), is a triple (A, f , B), denoted by f : A / / B, where f is an S-sorted mapping from A to B such that, for every (s, s) ∈ S × S, every σ ∈ Σ s,s and every (a j ) j∈|s| ∈ A s , we have f s (F σ ((a j ) j∈|s| )) = G σ ( f s ((a j ) j∈|s| )), where f s is the mapping ∏ j∈|s| f s j from A s to B s that sends (a j ) j∈|s| in A s to ( f s j (a j )) j∈|s| in B s . We will denote by Alg(Σ) the category of Σ-algebras and homomorphisms and by Alg(Σ) the set of objects of Alg(Σ).
In some cases, to avoid mistakes, we will denote by F A the structure of Σ-algebra on A, and, for (s, s) ∈ S × S and σ ∈ Σ s,s , by F A σ , or simply by σ A , the corresponding operation. Moreover, for s ∈ S and σ ∈ Σ λ,s , we will, usually, denote by σ A the value of the mapping F A σ : A λ / / A s at the unique element in A λ . We will denote by 1 S or, to abbreviate, by 1, the (standard) final Σ-algebra.
We add some further results concerning direct products and homomorphisms, which will be of great importance later on. Definition 15. Let (A i ) i∈I be a family of Σ-algebras, where we agree that, for every i ∈ I, The product of (A i ) i∈I , ∏ i∈I A i , is the Σ-algebra which has as S-sorted underlying set ∏ i∈I A i , and where, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , the structural operation F σ is defined as follows:

2.
For every i ∈ I, the i-th canonical projection is the homomorphism from ∏ i∈I A i to A i determined by the S-sorted mapping pr i which, for every s ∈ S, is defined as follows We next introduce the support of a many-sorted algebra and the definition of finiteness of a many-sorted algebra.
Definition 16. Let A be a Σ-algebra. Then the support of A, denoted by supp S (A), is supp S (A), i.e., the support of the underlying S-sorted set A of A.
Definition 17. Let A be a Σ-algebra. We will say that A is finite if A, the underlying S-sorted set of A, is finite.
We shall now go on to define the notion of subalgebra of a Σ-algebra A, the principle of proof by Algebraic Induction and the subalgebra generating operator for A. Definition 18. Let A = (A, F) be a Σ-algebra and X ⊆ A. Given (s, s) ∈ S × S and σ ∈ Σ s,s , we will say that X is closed under the operation F σ : A s / / A s if, for every (a j ) j∈|s| ∈ X s , F σ ((a j ) j∈|s| ) ∈ X s . We will say that X is a closed subset of A if X is closed under the operations of A. We will denote by Cl(A) the set of all closed subsets of A (which is an algebraic closure system on A) and by Cl(A) the algebraic lattice (Cl(A), ⊆). We will say that a Σ-algebra B is a subalgebra of A if B ⊆ A and the canonical embedding of B into A determines an embedding of B into A. We will denote by Sub(A) the set of all subalgebras of A. Since Cl(A) and Sub(A) are isomorphic, we shall feel free to deal either with a closed subset of A or with the correlated subalgebra of A, whichever is most convenient for the work at hand.
Definition 19. Let A be a Σ-algebra. Then we denote by Sg A the many-sorted closure operator on A defined as follows: We call Sg A the subalgebra generating many-sorted operator on A determined by A. For every X ⊆ A, we call Sg A (X) the subalgebra of A generated by X. Moreover, if X ⊆ A is such that Sg A (X) = A, then we say that X is an S-sorted set of generators of A, or that X generates A. Besides, we say that A is finitely generated if there exists an S-sorted subset X of A such that X generates A and card(X) < ℵ 0 .

Remark 4. Let
A be a Σ-algebra. Then the algebraic closure operator Sg A is uniform, i.e., for every X, Y ⊆ A, if supp S (X) = supp S (Y), then we have supp S (Sg A (X)) = supp S (Sg A (Y)).
We next recall the Principle of Proof by Algebraic Induction.
Proposition 7 (Principle of Proof by Algebraic Induction). Let A be a Σ-algebra generated by X. Then to prove that a subset Y of A is equal to A it suffices to show: 1.
Y is a subalgebra of A (algebraic induction step).

Proposition 8. Let
A be a Σ-algebra. Then the many-sorted closure operator Sg A on A is algebraic, i.e., for every S-sorted subset X of A, For a Σ-algebra A we next provide another, more constructive, description of the algebraic many-sorted closure operator Sg A .
Definition 20. Let Σ be an S-sorted signature and A a Σ-algebra.

1.
We denote by E A the many-sorted operator on A that assigns to an S-sorted subset X of A, where, for s ∈ S, Σ =λ,s is the set of all manysorted formal operations σ such that ar(σ) ∈ S − {λ}, car(σ) = s, and if ar(σ) = s, then X ar(σ) = ∏ i∈|s| X s i .

2.
If X ⊆ A, then we define the family (E n A (X)) n∈N in Sub(A), recursively, as follows:

3.
We denote by E ω A the many-sorted operator on A that assigns to an S-sorted subset X of A, E ω A (X) = n∈N E n A (X).

Remark 5.
Since, for s ∈ S and σ ∈ Σ λ,s , σ A is the value of F A σ : A λ / / A s at the unique element in A λ , it follows that Proposition 9. Let A be a Σ-algebra and X ⊆ A, then Sg A (X) = E ω A (X).

Remark 6.
For a homomorphism f from a Σ-algebra A to another B and a subset X of A, we have Our next goal is to define the concepts of congruence on a Σ-algebra and of quotient of a Σ-algebra by a congruence on it. Moreover, we recall the notion of kernel of a homomorphism between Σ-algebras, the universal property of the quotient of a Σ-algebra by a congruence on it and the congruence generating operator for A. Definition 21. Let A be a Σ-algebra and Φ an S-sorted equivalence on A. We will say that Φ is an S-sorted congruence on (or, to abbreviate, a congruence on) A if, for every (s, s) ∈ (S − {λ}) × S, every σ ∈ Σ s,s , and every (a j ) j∈|s| , (b j ) j∈|s| ∈ A s , if, for every j ∈ |s|, (a j , b j ) ∈ Φ s j , then (F σ ((a j ) j∈|s| ), F σ ((b j ) j∈|s| )) ∈ Φ s . We will denote by Cgr(A) the set of all S-sorted congruences on A (which is an algebraic closure system on A × A), by Cgr(A) the algebraic lattice (Cgr(A), ⊆), by ∇ A the greatest element of Cgr(A) and by ∆ A the least element of Cgr(A).
For a congruence Φ on A, the quotient Σ-algebra of A by Φ, denoted by A/Φ, is the Σ-algebra (A/Φ, F A/Φ ), where, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , the operation F A/Φ σ from (A/Φ) w to A s /Φ s , also denoted, to simplify, by F σ , sends ([a j ] Φ s j ) j∈|s| in (A/Φ) s to [F σ ((a j ) j∈|s| )] Φ s in A s /Φ s , and the canonical projection from A to A/Φ, denoted by pr Φ : A / / A/Φ, is the full surjective homomorphism determined by the projection from A to A/Φ. The ordered pair (A/Φ, pr Φ ) has the following universal property: Ker(pr Φ ) is Φ and, for every Σ-algebra B and every homomorphism f from A to B, if Ker( f ) ⊇ Φ, then there exists a unique homomorphism h from A/Φ to B such that h • pr Φ = f . In particular, if Ψ is a congruence on A such that Φ ⊆ Ψ, then we will denote by p Φ,Ψ the unique homomorphism from A/Φ to A/Ψ such that p Φ,Ψ • pr Φ = pr Ψ .

Remark 7.
Let ClfdAlg(Σ) be the category whose objects are the classified Σ-algebras, i.e, the ordered pairs (A, Φ) where A is a Σ-algebra and Φ a congruence on A, and in which the set of morphisms from (A, Φ) to (B, Ψ) is the set of all homomorphisms f from A to B such that, for every s ∈ S and every (x, y) ∈ A 2 s , if (x, y) ∈ Φ s , then ( f s (x), f s (y)) ∈ Ψ s . Let G be the functor from Alg(Σ) to ClfdAlg(Σ) whose object mapping sends A to (A, ∆ A ) and whose morphism mapping Then, for every classified Σ-algebra (A, Φ), there exists a universal mapping from (A, Φ) to G, which is precisely the ordered pair (A/Φ, pr Φ ) with Definition 22. Let A be a Σ-algebra. Then we denote by Cg A the many-sorted closure operator on A × A defined as follows: For a Σ-algebra A we next provide another, more constructive, description of the algebraic many-sorted closure operator Cg A .
Definition 23. Let Σ be an S-sorted signature and A a Σ-algebra.
(1) We denote by C A the many-sorted operator on A × A that assigns to an S-sorted relation Φ ⊆ A × A, the S-sorted relation where, for s ∈ S, Σ =λ,s is the set of all many-sorted formal operations σ such that ar(σ) ∈ S − {λ}, car(σ) = s and if ar(σ) = s, then Φ ar(σ) = ∏ j∈|s| Φ s j . Let us note that (2) If Φ ⊆ A × A, then we define the family (C n A (Φ)) n∈N in Sub(A × A), recursively, as follows: (3) We denote by C ω A the many-sorted operator on A × A that assigns to an S-sorted relation Proposition 10. Let A be a Σ-algebra and let Φ ⊆ A × A be a relation on A. For every n ∈ N we have that (1) C n A (Φ) is a reflexive relation; We next state that the forgetful functor G Σ from Alg(Σ) to Set S has a left adjoint T Σ which assigns to an S-sorted set X the free Σ-algebra T Σ (X) on X. We latter state the universal property of the free many-sorted algebra and provide a characterization by means of the notion of Dedekind Peano-algebras.
Let us note that in what follows, to construct the algebra of Σ-rows in X, and the free Σ-algebra on X, since neither the S-sorted signature Σ nor the S-sorted set X are subject to any constraint, coproducts must necessarily be used.
Definition 24. Let X be an S-sorted set. The algebra of Σ-rows in X, denoted by W Σ (X), is defined as follows: 1.
The underlying S-sorted set of W Σ (X), written as W Σ (X), is precisely the S-sorted set (( Σ X) ) s∈S , i.e., the mapping from S to U constantly ( Σ X) , where ( Σ X) is the set of all words on the set Σ X, i.e., on the set

Definition 25.
The free Σ-algebra on an S-sorted set X, denoted by where, for every s ∈ S and every x ∈ X s , (x) stands for (((x, s), 1)), which is the value at x of the canonical mapping from X s to ( Σ X) . We will denote by T Σ (X) the underlying S-sorted of T Σ (X) and, for s ∈ S, we will call the elements of T Σ (X) s terms of type s with variables in X or (X, s)-terms. Remark 8. Since ({(x) | x ∈ X s }) s∈S is a generating subset of T Σ (X), to prove that a subset T of T Σ (X) is equal to T Σ (X) it suffices, by Proposition 21, to show: 1.
T is a subalgebra of T Σ (X) (algebraic induction step).
In the many-sorted case we have, as in the single-sorted case, the following characterization of the elements of T Σ (X) s , for s ∈ S. Proposition 12. Let X be an S-sorted set. Then, for every s ∈ S and every P ∈ W Σ (X) s , we have that P is a term of type s with variables in X if and only if P = (x), for a unique x ∈ X s , or P = (σ), for a unique σ ∈ Σ λ,s , or P = (σ) (P j ) j|s| , for a unique s ∈ S − {λ}, a unique σ ∈ Σ s,s and a unique family (P j ) j∈|s| ∈ T Σ (X) s . Moreover, the three possibilities are mutually exclusive. From now on, for simplicity of notation, we will write x, σ and σ(P 0 , . . . , P |s|−1 ) or σ((P j ) j∈|s| ) instead of (x), (σ) and (σ) (P j ) j∈|s| , respectively. Thus, in particular, the structural operation F σ (or, more accurately, F T Σ (X) σ ) associated to σ is identified with σ.
From the above proposition it follows, immediately, the universal property of the free Σ-algebra on an S-sorted set X, as stated in the subsequent proposition.

Proposition 13.
For every S-sorted set X, the pair (η X , T Σ (X)), where η X , the insertion of (the S-sorted set of generators) X into T Σ (X), is the co-restriction to T Σ (X) of the canonical embedding of X into W Σ (X), has the following universal property: for every Σ-algebra A and every S-sorted mapping f : X / / A, there exists a unique homomorphism f : Proof. For every s ∈ S and every (X, s)-term P, the s-th coordinate f s of f is defined recursively as follows: The just stated proposition allows us to carry out definitions by algebraic recursion on a free many-sorted algebra as indeed we will be doing throughout this paper.
Corollary 1. The functor T Σ , which sends an S-sorted set X to T Σ (X) and an S-sorted mapping It is possible to give an internal characterization of the free algebras by means of the Dedekind-Peano algebras.
Definition 26. Let A be a Σ-algebra. We will say that A is a Dedekind-Peano Σ-algebra, abbreviated to DP Σ-algebra when this is unlikely to cause confusion, if the following axioms hold DP1. For every (s, s) ∈ S × S and every σ ∈ Σ s,s , DP2. For every s ∈ S and every σ, We call the S-sorted set A − ( σ∈Σ ·,s Im(F σ )) s∈S the basis of Dedekind-Peano of A, and we denote it by B(A).
We remark that the axioms DP1 and DP2 are equivalent to the following axiom Proposition 15. Let X be an S-sorted set. Then T Σ (X) is a DP Σ-algebra.

Many-sorted partial algebras
In this section we gather together the basic facts about many-sorted partial algebras that we will need afterwards. Specifically, we define the notion of many-sorted partial algebra, we introduce the homomorphisms between many-sorted partial algebra and several classes of them, as e.g., the closed and the full homomorphisms. Moreover, we define, in connection with the homomorphisms and their properties, several classes of subobjects and quotient objects of the many-sorted partial algebras.
is called a discrete operation or an empty (partial) operation from A s to A s . If (s, s) = (λ, s), σ ∈ Σ λ,s and Dom(F λ,s (σ)) = ∅, then F λ,s (σ) picks out an element of A s . Therefore a partial operation F λ,s (σ) from A λ to A s , also called a partial constant F λ,s (σ) from A λ to A s , is either discrete or distinguishes exactly an element of A s . For a pair (s, s) ∈ S × S and a formal operation σ ∈ Σ s,s , in order to simplify the notation, the partial operation F s,s (σ) from A s to A s will be written as F σ . A many-sorted partial Σ-algebra (or, to abbreviate, partial Σ-algebra) is a pair (A, F), denoted by A, where A is an S-sorted set and F a structure of partial Σ-algebra on A. A Σ-homomorphism (or, to abbreviate, homomorphism) from A to B, where B = (B, G), is a triple (A, f , B), denoted by f : A / / B, where f is an S-sorted mapping from A to B such that, for every (s, s) ∈ S × S, every σ ∈ Σ s,s and every (a j ) j∈|s| ∈ A s , if (a j ) j∈|s| ∈ Dom(F σ ), then ( f s j (a j )) j∈|s| ∈ Dom(G σ ) and f s (F σ ((a j ) j∈|s| )) = G σ ( f s ((a j ) j∈|s| )). We will denote by Preprints (www.preprints.org) | NOT PEER-REVIEWED | Posted: 27 September 2021 doi:10.20944/preprints202109.0445.v1 PAlg(Σ) the category of partial Σ-algebras and homomorphisms and by PAlg(Σ) the set of objects of PAlg(Σ).

Remark 9.
The forgetful functor from the category PAlg(Σ) to the category Set S has a left adjoint, denoted by D Σ , which sends an S-sorted set A to D Σ (A), the discrete partial Σ-algebra on A, defined as follows: Its underlying S-sorted set is A, and, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , Dom(F σ ) = ∅.

Remark 10.
The category Alg(Σ) is a full subcategory of the category PAlg(Σ). We shall prove later that the canonical full embedding of Alg(Σ) into PAlg(Σ) has a left adjoint, the free completion of a partial Σ-algebra, which will play a key role in this paper.
The following is a useful characterization of the isomorphisms (that is (partially) analogous to that of the homeomorphisms, i.e., bijective and bicontinuous applications between topological spaces). Later we will include additional characterizations of the isomorphisms by using the notions of closed homomorphism and of full homomorphism.
Proposition 16. Let f be a homomorphism from A to B. Then the following statements are equivalent: 1.
f is an isomorphism from A to B, i.e., f is a section and a retraction.

f is bijective and
To establish some of the results that follow, it is useful to have the concept of product of a family of many-sorted partial algebras.
Definition 28. Let (A i ) i∈I be a family of partial Σ-algebras, where we agree that, for every i ∈ I, The product of (A i ) i∈I , ∏ i∈I A i , is the partial Σ-algebra which has as S-sorted underlying set ∏ i∈I A i , and where, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , the structural operation F σ is defined as follows: For every i ∈ I, the i-th canonical projection is the homomorphism from ∏ i∈I A i to A i determined by the S-sorted mapping pr i which, for every s ∈ S, is defined as follows In what follows we define the notions of subalgebra, relative subalgebra and weak subalgebra of a partial Σ-algebra A and the subalgebra generating operator for A. Furthermore, we characterize the subalgebras, the relative subalgebras and the weak subalgebras by means of the closed homomorphisms, the full homomorphisms and the homomorphisms, respectively.
Definition 29. Let A be a partial Σ-algebra and X ⊆ A. Given (s, s) ∈ S × S and σ ∈ Σ s,s , we will say that X is closed under the partial operation We will say that X is a closed subset of A if X is closed under the partial operations of A. We will denote by Cl(A) the set of all closed subsets of A (which is an algebraic closure system on A) and by Cl(A) the algebraic lattice (Cl(A), ⊆). We will say that a partial Σ-algebra . We will denote by Sub(A) the set of all subalgebras of A and by Sub(A) the ordered set (Sub(A), ⊆). Since Cl(A) and Sub(A) are isomorphic, we shall feel free to deal either with a closed subset of A or with the correlated subalgebra of A, whichever is most convenient for the work at hand.

Remark 12.
Let A and B be partial Σ-algebras such that B ⊆ A. Then B = (B, G) is a subalgebra of A if and only if, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , Remark 13. Let A be a partial Σ-algebra. If, for every s ∈ S and every σ ∈ Σ λ,s , Dom(F σ ) = ∅, then ∅ S ∈ Cl(A). Let us note that, in such a case, ∅ ∅ ∅ S , the subalgebra of A canonically associated to ∅ S , is such that, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , the partial operation from ∅ S s to ∅ S s = ∅ associated to σ is discrete.
We next define the notion of closed homomorphism from a partial Σ-algebra to another which will allow us, among other things, to provide another characterization of the isomorphisms and of the subalgebras of a partial Σ-algebra.
Definition 30. Let A and B be partial Σ-algebras and f a homomorphism from A to B. We will say that f is closed if, for every (s, s) ∈ S × S, every σ ∈ Σ s,s and every (a

Remark 14.
A homomorphism is closed if and only if the domains of the partial operations in the source are exactly the inverse images of the domains of the corresponding operations in the target.

Remark 15.
Every homomorphism between Σ-algebras is a closed homomorphism.
Proposition 18. Let f be a homomorphism from A to B. Then the following statements are equivalent: 1. f is an isomorphism from A to B, i.e., f is a section and a retraction. 2.
f is bijective and closed, i.e., for every (s, s) ∈ S × S, every σ ∈ Σ s,s and every (a j ) j∈|s| ∈ A s , the following holds: Proposition 19. Let A and B be partial Σ-algebras, f a homomorphism from A to B and X and Y closed subsets of A and B, respectively. Then: If f is closed, then f [X] is a closed subset of B.

Remark 16.
For a homomorphism f from a partial Σ-algebra A to another B and a subset X of A, Definition 31. Let A be a partial Σ-algebra. Then we will denote by Sg A the algebraic closure operator canonically associated to the algebraic closure system Cl(A) on A and we call it the subalgebra generating operator for A. Moreover, if X ⊆ A, then we call Sg A (X) the subalgebra of A generated by X, and if X is such that Sg A (X) = A, then we will say that X is a generating subset of A. Besides, Sg A (X) denotes the partial Σ-algebra determined by Sg A (X).

Remark 17.
Let A be a partial Σ-algebra. Then the algebraic closure operator Sg A is uniform, i.e., for every X, Proposition 21 (Principle of Proof by Algebraic Induction). Let A be a partial Σ-algebra generated by X. Then to prove that a subset Y of A is equal to A it suffices to show: 1. X ⊆ Y (algebraic induction basis); and 2.
Y is a closed subset subalgebra of A (algebraic induction step).
We next state the principle of extension of identities. This principle, which is fundamental to elucidate the equality of two coterminal homomorphisms, will be used on several occasions in this work.
Proposition 22. Let f , g : A / / B be homomorphisms between partial Σ-algebras and let X be a subset of A. If f X = g X , then f Sg A (X) = g Sg A (X) . In particular, if X is a generating subset of A and f X = g X , then f = g Definition 32. Let A be a partial Σ-algebra. We will say that a partial Σ-algebra B = (B, G) is a relative subalgebra of A if B ⊆ A and, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , We will denote by Sub r (A) the set of all relative subalgebras of A.
Remark 18. Let A and B be partial Σ-algebras such that B ⊆ A. Then B = (B, G) is a relative subalgebra of A if and only if, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , Remark 19. Let A = (A, F) be a partial Σ-algebra and X ⊆ A. Then there exists a unique structure of partial Σ-algebra L in X,A (F) on X, the optimal lift of F with respect to in X,A , such that, for every partial Σ-algebra C and every mapping h from C to X, if in X,A • h is a homomorphism from C to A, then h is a homomorphism from C to (X, L in X,A (F)). In fact, it suffices to take, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , as partial operation associated to σ the partial mapping L in X,A (F) σ from X s to X s defined as follows: . This shows that every subset X of A is the underlying S-sorted set of exactly one relative subalgebra of A, and that the structure of partial Σ-algebra on X is uniquely determined by the structure of partial Σ-algebra on A and the specification of the subset X of A.
We next define the notion of full homomorphism from a partial Σ-algebra to another which will allow us, among other things, to provide another characterization of the isomorphisms and of the relative subalgebras of a Σ-algebra.
Definition 33. Let A and B be partial Σ-algebras and f a homomorphism from A to B. We will say that f is full if, for every (s, s) ∈ S × S, every σ ∈ Σ s,s and every (a Remark 20. Those homomorphisms that are full and injective are characterizable as those that are initial and injective, where a homomorphism f from a partial Σ-algebra A to another B is said to be initial if, for every partial Σ-algebra C and every S-sorted mapping h from C to A, if f • h is a homomorphism from C to B, then h is a homomorphism from C to A.
Proposition 23. Let f be a homomorphism from A to B. Then the following statements are equivalent: 1. f is an isomorphism from A to B, i.e., f is a section and a retraction. 2.
f is bijective and full.
Proposition 24. Let A and B be partial Definition 34. Let A be a partial Σ-algebra. We will say that a partial Σ-algebra Remark 22. Since every closed homomorphism is a full homomorphism and every full homomorphism is a homomorphism, it follows that every subalgebra is a relative subalgebra and every relative subalgebra is a weak subalgebra. Moreover, every relative subalgebra on a closed subset is a subalgebra.
Proposition 26. Let A and B be partial Σ-algebras and f : A / / B a mapping from A to B. Then: 1.

2.
If B is a Σ-algebra and Γ f is a closed subset of A × B, then f is a homomorphism from A to B.
Remark 23. Notice the (partial) analogy of the just stated proposition and the closed graph theorem in point-set topology.
Lemma 1. Let f : A / / B be a homomorphism, X ⊆ A, g : X / / B such that Γ g ⊆ Γ f and X the relative subalgebra of A on X. Then: Corollary 2. Let f : A / / B be a homomorphism and X ⊆ A. Then the following statements are equivalent: The following technical lemma will become quite useful later on.
Lemma 2. Let A, B, C and D be partial Σ-algebras, C a weak subalgebra of A such that Sg A (C) = A, D a relative subalgebra of B and f a homomorphism from C to D which allows a homomorphic extension f from A to B. Then: is the relative subalgebra of A whose underlying many-sorted set is Dom(Sg A×D (Γ f ))) and Dom(Sg A×D (Γ f )) is generated by C.

2.
If E is a C-generated relative subalgebra of A and g : E / / D is a homomorphic extension of f , then Γ g ⊆ Γ f .

3.
If A is a Σ-algebra, then f is a closed homomorphism from Dom(Sg A×D (Γ f )) to D; and if f is closed and g : E / / D is any closed homomorphic extension of f such that E is a C-generated relative subalgebra of A, then g = f .
Definition 35. Let A and B be partial Σ-algebras and f a homomorphism from A to B. We will Remark 24. If a homomorphism between partial Σ-algebras is dense and closed, then it is surjective (because the image of a closed homomorphism is a closed subset of the target).
Proposition 27. Let A and B be partial Σ-algebras and f a homomorphism from A to B. Then there exists a unique epimorphism f e from A to and only if f is closed and injective; and (2) for every partial Σ-algebra C, every epimorphism g from A to C and every closed injective Therefore (Epimorphisms, Closed and injective homomorphisms) is a factorization system in PAlg(Σ).
Congruences provide an internal description of the homomorphic images from a partial Σ-algebra and closed congruences provide an internal description of the closed homomorphic images from a partial Σ-algebra. Moreover, closed congruences are fundamental for the description of the varieties of partial Σ-algebras defined by existentially conditional existence equations.
Definition 36. Let A be a partial Σ-algebra and Φ an S-sorted equivalence on A. We will say that Φ is an S-sorted congruence on (or, to abbreviate, a congruence on) A if, for every (s, s) ∈ (S − {λ}) × S, every σ ∈ Σ s,s , and every (a j ) j∈|s| , (b j ) j∈|s| ∈ A s , if, for every j ∈ |s|, (a j , b j ) ∈ Φ s j , (a j ) j∈|s| ∈ Dom(F σ ) and (b j ) j∈|s| ∈ Dom(F σ ), then (F σ ((a j ) j∈|s| ), F σ ((b j ) j∈|s| )) ∈ Φ s . We will denote by Cgr(A) the set of all S-sorted congruences on A, which is an algebraic closure system on A × A, by Cg A the corresponding algebraic closure operator, by Cgr(A) the algebraic lattice (Cgr(A), ⊆), by ∇ A the greatest element of Cgr(A) and by ∆ A the least element of Cgr(A).
For a congruence Φ on A, the quotient partial Σ-algebra of A by Φ, denoted by A/Φ, is the partial Σ-algebra (A/Φ, F A/Φ ), where, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , the domain of the partial operation F A/Φ σ , also denoted, to simplify, by F σ , from (A/Φ) s to A s /Φ s is the set to B such that h • pr Φ = f . In particular, if Ψ is a congruence on A such that Φ ⊆ Ψ, then we will denote by p Φ,Ψ the unique homomorphism from A/Φ to A/Ψ such that p Φ,Ψ • pr Φ = pr Ψ . We will say that Φ is an S-sorted closed congruence on (or, to abbreviate, a closed congruence on) A if Φ is a congruence on A and, for every (s, s) ∈ (S − {λ}) × S, every σ ∈ Σ s,s , and every (a j ) j∈|s| , (b j ) j∈|s| ∈ A s , if, for every j ∈ |s|, (a j , b j ) ∈ Φ s j and (a j ) j∈|s| ∈ Dom(F σ ), then (b j ) j∈|s| ∈ Dom(F σ ). We will denote by Cgr c (A) the set of all S-sorted closed congruences on A. The congruence ∆ A is a closed congruence on A, while ∇ A is closed if and only if, for every (s, s) ∈ (S × S and every σ ∈ Σ s,s , F σ is either total or discrete. If Φ and Ψ are closed congruences on A, then Cg A (Φ ∪ Ψ), their supremum in Cgr(A), is a closed congruence on A and it is the least equivalence relation containing them, i.e., Cg A (Φ ∪ Ψ) = n∈N (Ψ • Φ) n (let us point out that, in general, the supremum of two non-closed congruences is not necessarily the least equivalence relation containing them). The set Cgr c (A) is a principal ideal of Cgr(A) precisely the one determined by Φ∈Cgr c (A) Φ, which is the largest closed congruence on A.
Remark 25. Let A be a partial Σ-algebra and Φ an S-sorted equivalence on A. Then Φ is a congruence on A if and only if Φ is a subalgebra of A × A.
Remark 26. Let A be a partial Σ-algebra and Φ an S-sorted congruence on A, then the full and surjective homomorphism pr Φ from A to A/Φ is final and surjective, where a homomorphism f from a partial Σ-algebra A to another B is said to be final if, for every partial Σ-algebra C and every S-sorted mapping h from B to C, if h • f is a homomorphism from A to C, then h is a homomorphism from B to C. For a homomorphism f from a partial Σ-algebra A to another B the property of being final is very interesting because it allows one to state that an S-sorted mapping h from B to C, the underlying set of another partial Σ-algebra C, is a homomorphism if its composition with the subjacent application of the homomorphism f is a homomorphism from A to C. Let us point out that a homomorphism f from a partial Σ-algebra A = (A, F) to another B = (B, G) is final if and only if, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , Remark 27. Let A be a partial Σ-algebra. If Φ an S-sorted closed congruence on A, then pr Φ : A / / A/Φ, is a closed and surjective homomorphism.
Remark 28. Let ClfdPAlg(Σ) be the category whose objects are the classified partial Σ-algebras, i.e, the ordered pairs (A, Φ) where A is a partial Σ-algebra and Φ a congruence on A, and in which the set of morphisms from (A, Φ) to (B, Ψ) is the set of all homomorphisms f from A to B such that, for every s ∈ S and every (x, y) ∈ A 2 s , if (x, y) ∈ Φ s , then ( f s (x), f s (y)) ∈ Ψ s . Let G be the functor from PAlg(Σ) to ClfdPAlg(Σ) whose object mapping sends A to (A, ∆ A ) and whose morphism mapping sends f : A / / B to f : (A, ∆ A ) / / (B, ∆ B ). Then, for every classified partial Σ-algebra (A, Φ), there exists a universal mapping from (A, Φ) to G, which is precisely the Remark 29. Let A = (A, F) be a partial Σ-algebra and Φ an S-sorted equivalence on A. Then there exists a unique structure of partial Σ-algebra L pr Φ (F) on A/Φ, the co-optimal lift of F with respect to pr Φ , such that, for every partial Σ-algebra C and every mapping h from A/Φ to C, if h • pr Φ is a homomorphism from A to, C then h is a homomorphism from C to (A/Φ, L pr Φ (F)) if and only if Φ is an S-sorted congruence on A. This shows that, for every S-sorted congruence Φ on A A, A/Φ is the underlying S-sorted set of exactly one Σ-algebra, and that the structure of partial Σ-algebra on A/Φ is uniquely determined by the structure of partial Σ-algebra on A and the specification of the S-sorted congruence Φ on A. What we have just said is, in fact, a particular case of the following: for a partial Σ-algebra A = (A, F) and an S-sorted mapping f from A to B, there exists a co-optimal lift of F with respect to f if and only if Ker( f ) is a congruence on A.
We next state the weak homomorphism theorem.
Proposition 28. Let A, B and C be partial Σ-algebras, f a full and surjective homomorphism from A to B and g a homomorphism from A to C. If Ker(g) ⊇ Ker( f ), then there exists a unique h is an isomorphism if and only if g is a surjective and full homomorphism and Ker(g) = Ker( f ); 3.
if g is closed, then h is closed; 4.
if f is closed, then h is closed if and only if g is closed.
From the just stated proposition we obtain the weak isomorphism theorem. for every partial Σ-algebra C, every full and surjective homomorphism g from A to C and every injective homomorphism h from C to B, if f = h • g, then there exists a unique isomorphism k from C to A/Ker( f ) such that k • g = pr Ker( f ) and f m • k = h.

Partial Dedekind-Peano Algebras
We next define for many-sorted partial algebras the counterpart of the notion of Dedekind-Peano algebra. Moreover, we state and prove the principle of definition by algebraic recursion for free many-sorted algebras with respect to many-sorted partial algebras and define a functor from a wide subcategory of PAlg(Σ) to Alg(Σ) which will be used afterwards to obtain, from a subfunctor of that functor, an example of completion.
Definition 37. Let A be a partial Σ-algebra. We will say that A is a partial Dedekind-Peano Σ-algebra, abbreviated to PDP-algebra when this is unlikely to cause confusion, if the following axioms hold PDP1. For every (s, s) ∈ S × S and every σ ∈ Σ s,s , F σ : A s / A s is injective, i.e., for every For every s ∈ S and every σ, τ ∈ Σ ·,s , if σ = τ, then Im(F σ ) ∩ Im(F τ ) = ∅, i.e., We call the S-sorted set A − ( σ∈Σ ·,s Im(F σ )) s∈S the basis of Dedekind-Peano of A, and we denote it by B(A).
We next state and prove the principle of definition by algebraic recursion for free many-sorted Σ-algebras, i.e., many-sorted DP-algebras, with respect to many-sorted partial Σ-algebras, which, for many-sorted partial algebras, is more important than the principle of definition by algebraic recursion for PDP-algebras with respect to many-sorted algebras.
Proposition 31. Let X be an S-sorted set, A a partial Σ-algebra and f an S-sorted mapping from X to A. Then there exists a unique homomorphism f ∂ is a closed homomorphism. 5.
f ∂ is the largest homomorphic extension of f to an X-generated relative subalgebra of T Σ (X) with codomain A.
Proof. Let A ∞ be the Σ-algebra defined as follows: The underlying S-sorted set of A ∞ is (A s ∪ {A s }) s∈S and, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , the operation We will call A ∞ the one-point per sort completion of the partial Σ-algebra A. Notice that A is a relative subalgebra of A ∞ , that A ∞ need not be generated by A (since some of the new elements are not necessarily accessible from A) and that the canonical embedding in A,A ∞ of A into A ∞ is the underlying mapping of a full and injective homomorphism, denoted by in A,A ∞ , from A to A ∞ . Then, by the universal property of the free algebra, there exists a unique homomorphism Then ∂( f ), the relative subalgebra of T Σ (X) on ∂( f ), together with the homomorphism determined by f ∂ , which, with the customary abuse of notation, we denote by the same symbol, satisfy the desired conditions.
To better understand the mappings at play, we provide the commutative diagram in Figure 3.

Remark 31.
Let us note that, by the Axiom of Regularity, we have that, for every s ∈ S, A s ∩ {A s } = ∅. Moreover, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , Γ , the partial operation from ∂( f ) s to ∂( f ) s associated to σ, is On the other hand, Γ f ∂ = Sg T Σ (X)×A ((η X × id A )[Γ f ]), i.e., the underlying function of f ∂ , is the subalgebra of the partial Σ-algebra T Σ (X) × A generated by the image under the S-sorted mapping η X × id A : X × A / / T Σ (X) × A, of the underlying function of f . Finally, we have that 1.
For every s ∈ S and every x ∈ X s , f ∂ s (x) = f s (x).

3.
For every (s, s) ∈ (S − {λ}) × S, every σ ∈ Σ s,s and every (P j ) j∈|s| ∈ T Σ (X) s , if, for every j ∈ |s|, P j ∈ ∂( f ) s j and the family ( The construction that assigns to a partial Σ-algebra A the Σ-algebra A ∞ is not the object mapping of a functor left adjoint to the inclusion functor In Alg(Σ) from Alg(Σ) to PAlg(Σ). However, the just mentioned construction is the object mapping of a functor (·) ∞ from the category PAlg(Σ) c , of many-sorted partial Σ-algebras and closed homomorphisms, to the category Alg(Σ). For later use, specifically, when we consider completions, we next prove such a result.
Proposition 32. There exists a functor (·) ∞ from the category PAlg(Σ) c of many-sorted partial Σ-algebras and closed homomorphisms to the category Alg(Σ).
Proof. Let A = (A, F) and B = (B, G) be many-sorted partial Σ-algebras and f : A / / B a closed homomorphism. Then the many-sorted mapping f ∞ from A ∞ to B ∞ defined, for every sort s ∈ S, as follows is the underlying mapping of a homomorphism, also denoted by f ∞ , which we will call the one-point per sort completion of f , from A to B. In fact, let (s, s) be a pair in S × S, σ an operation symbol in Σ s,s and (z j ) j∈|s| a family of elements in A ∞ s . Then it could be the case that either (1) (z j ) j∈|s| ∈ Dom(F A σ ), or (2) (z j ) j∈|s| ∈ Dom(F A σ ). In Case (1), i.e., if (z j ) j∈|s| ∈ Dom(F σ ), then we have that f s ((z j ) j∈|s| ) ∈ Dom(G σ ). Moreover, the following chain of equalities holds In the just stated chain of equalities, the first equality follows from the definition of F ∞ σ , as stated in the proof of Proposition 31; the second equality follows from the fact that, since (z j ) j∈|s| ∈ Dom(F σ ), F σ ((z j ) j∈|s| ) is an element of A s and the image under f ∞ s of this last element is identical to its image under f s ; the third equality follows from the fact that, by assumption, f is a homomorphism from A to B; the fourth equality follows from the fact that (z j ) j∈|s| ∈ Dom(F σ ), therefore the image under f ∞ s of this last family coincides with its image under f s ; finally, the last equality follows from the fact that f s ((z j ) j∈|s| ) ∈ Dom(G σ ), f s ((z j ) j∈|s| ) = f ∞ s ((z j ) j∈|s| ) and the definition of G ∞ σ , as stated in the proof of Proposition 31.
In Case (2), i.e., if (z j ) j∈|s| ∈ Dom(F σ ), then, since f is a closed homomorphism we have, by Definition 30, that f s ((z j ) j∈|s| ) ∈ Dom(G σ ). Moreover, the following chain of equalities holds The first equality follows from the definition of F ∞ σ , as stated in the proof of Proposition 31, and the fact that we are assuming that (z j ) j∈|s| ∈ Dom(F A σ ); the second equality follows from the definition of f ∞ ; the third equality follows from the fact that f s ((z j ) j∈|s| ) ∈ Dom(G σ ); for the fourth equality, since f s ( Therefore f ∞ is a homomorphism from A ∞ to B ∞ . Let us notice that the property of being closed of f has been crucial for the proof of Case (2) and cannot be weakened.
We let (·) ∞ stand for the pair of mappings that assign 1.
to a partial Σ-algebra A, the Σ-algebra A ∞ , as defined in the proof of Proposition 31; and 2.
to a closed homomorphism f : We next prove that the (·) ∞ is a functor from PAlg(Σ) c to Alg(Σ). Let A be a partial Σ-algebra. We want to prove that i.e., that the one-point per sort completion (id A ) ∞ : Let s be a sort in S and z an element in A ∞ s . It could be the case that either (1) z ∈ A s or (2) z = A s .
In Case (1), i.e., if z ∈ A s , then we have that The first equality follows from the fact that z ∈ A s and the definition of (id A ) ∞ s (z); the second equality gives the value of the mapping (id A ) s at the element z ∈ A s ; and, finally, the last equality recovers the value of the mapping (id A ∞ ) s at the element z ∈ A ∞ s .
In Case (2), i.e., if z = A s , then we have that The first equality follows from the fact that z = A s ; the second equality follows from the definition of (id A ) ∞ s (A s ); and, finally, the last equality recovers the value of the mapping (id A ∞ ) s at the element z = A s ∈ A ∞ s . In any case, we conclude that (id A ) ∞ = id A ∞ . Now let f : A / / B and g : B / / C be two closed homomorphisms. Consider the one-point per sort completions f ∞ : A ∞ / / B ∞ and g ∞ : Let s be a sort in S and z an element in A ∞ s . It could be the case that either (1) z ∈ A s or (2) z = A s .
In Case (1), i.e., if z ∈ A s , then we have that The first equality follows from the fact that z ∈ A s and the definition of (g • f ) ∞ s (z); the second equality unravels the s-sorted component of the composition of two many-sorted mappings; the third equality follows from the fact that z ∈ A s , f s (z) ∈ B s and the definitions of f ∞ s (z) and g ∞ s ( f ∞ s (z)); finally, the last equality recovers the s-sorted component of the composition of two many-sorted mappings.
In Case (2), i.e., if z = A s , then the following chain of equalities holds The first equality follows from the fact that z = A s ; the second equality follows from the definition of (g • f ) ∞ s (A s ); the third equality follows from the fact that f ∞ s (A s ) = B s and the definitions of f ∞ s (A s ) and g ∞ s ( f ∞ s (A s )); finally, the last equality recovers the s-sorted component of the composition of two many-sorted mappings.
In any case, we conclude that (g All in all, we conclude that (·) ∞ is a functor from PAlg(Σ) c to Alg(Σ).

The free completion of a many-sorted partial Σ-algebra and the Schmidt construction
We next prove, as a consequence of a well-known theorem about adjoint functors, that the inclusion functor In Alg(Σ) from Alg(Σ) to PAlg(Σ) has a left adjoint, the (absolutely) free completion functor. We point out that the free completion of a many-sorted partial Σ-algebra is one of the most useful tools of the theory of partial algebras. Later on, once we have defined the notion of completion and a suitable category Cmpl(Σ), of completions, we will prove that the free completion is a weakly initial object of Cmpl(Σ). At the end of this section, and related to the free completion functor, we state the generalized recursion theorem of Schmidt (see [17]), which we will also call the Schmidt construction. This construction, which, as stated in the introduction, is fundamental in the field of many- sorted partial algebras, will be treated later from a more general point of view and, in addition, will be functorialized. Figure 4 commutes and the following conditions are satisfied:

1.
A is complete, well-powered and co-well-powered, 2.
U has a left-adjoint, 4.
V is faithful, then G has a left adjoint. Figure 5 commutes and 1. Alg(Σ) is complete, well-powered and co-well-powered, 2.
Therefore In Alg(Σ) has a left adjoint F Σ , the free completion functor.
We next provide an explicit construction of the free completion of a partial Σ-algebra.
Proposition 34. Let A be a partial Σ-algebra. Then there exists a Σ-algebra F Σ (A), the free completion of A, and a homomorphism η A from A to F Σ (A) such that, for every Σ-algebra B and every homomorphism f from A to B, there exists a unique homomorphism f fc from Proof. Let T Σ (A) be the free Σ-algebra on A and, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , let F σ be the mapping from T Σ (A) s to T Σ (A) s defined as: Thus F σ has been obtained by activating F A σ , the partial operation of A associated to σ, when this is possible, and when this is not the case, retrieving a syntactic term. Let generated by A, together with η A , the canonical embedding of A into F Σ (A) induced by the corestriction of the mapping η A : A / / T Σ (A) to the underlying S-sorted set of F Σ (A), is a universal morphism from A to In Alg(Σ) . In fact, from F Σ (A) we obtain a partial Dedekind-Peano Σ-algebra F * Σ (A) = (F Σ (A), F * ) by defining, for every (s, s) ∈ S × S and every σ ∈ Σ s,s , F * σ to be the partial mapping from F Σ (A) s to F Σ (A) s whose domain of definition is F Σ (A) s − Dom(F A σ ). Then, by Proposition 30, there exists a unique homomorphism Then it suffices to take as f fc the homomorphism f but considered as a homomorphism from F Σ (A) to B. Corollary 5. The functor F Σ from PAlg(Σ) to Alg(Σ), which sends a partial Σ-algebra A to F Σ (A) and a homomorphism f from A to B to f @ , the unique homomorphism Remark 32. For every partial Σ-algebra A, the injective homomorphism η A from A to F Σ (A), i.e., the value of the unit of the adjuction at A, is, in addition, an epimorphism, hence a bimorphism. Therefore, the full subcategory Alg(Σ) of PAlg(Σ) is monoreflective and epireflective.
We shall let α A stand for (in A,A ∞ ) fc . To show that α = (α A ) A∈PAlg(Σ) c is a natural transformation from F Σ to (·) ∞ we need to check that, for every closed homomorphism f : i.e., that the following diagram From the left hand side of ( †) we obtain the following chain of equalities In the just stated chain of equalities the first equality follows by the associativity of the composition of S-sorted mappings; the second equality follows from Corollary 5; the third equality follows by the associativity of the composition of S-sorted mappings; and, finally, the last equality follows from the universal property of the free completion stated in Proposition 34 for the homomorphism in B,B ∞ . From the right hand side of ( †) we obtain the following chain of equalities In the just stated chain of equalities the first equality follows by the associativity of the composition of S-sorted mappings and the last equality follows from the universal property of the free completion stated in Proposition 34 for the homomorphism in A,A ∞ . Therefore, in order to check that ( †) holds it suffices to check that Let s be a sort in S and a ∈ A s , then we have the following chain of equalities  Remark 33. Let X be an S-sorted set. Then, for D Σ (X), the discrete many-sorted partial Σ-algebra associated to an S-sorted set X, the three many-sorted Σ-algebras F Σ (D Σ (X)), T Σ (D Σ (X)) and T Σ (D Σ (X)) are equal to T Σ (X), the free many-sorted Σ-algebra on X. Moreover, for every manysorted partial Σ-algebra B, we have that every mapping f from X to the underlying S-sorted set of B is a homomorphism f from D Σ (X) to B.
To better understand the working of the functor F Σ , we provide the commutative diagram in Figure 6. Let us note that for f : D Σ (X) / / B the diagram in Figure 6 becomes the commutative diagram in Figure 7.
Remark 34. If A is a Σ-algebra, then A together with id A is a free completion of A. Moreover, for every S-sorted set X, T Σ (X) together with η X is a free completion of D Σ (X).
In the following proposition we state that the free completion of a partial Σ-algebra can also be characterized internally.
Proposition 36. Let A be a partial Σ-algebra and B a Σ-algebra such that A is a weak subalgebra of B. Then B together with in A,B , the canonical embedding of A into B, is a free completion of A if, and only if, the following conditions hold: FC1. For every (s, s) ∈ S × S, every σ ∈ Σ s,s and every ). This condition, as Burmeister, in [17] on p. 80, says "means that no value of a fundamental operation which lies in A 'can come from the outside' ". FC2. For every s ∈ S, every σ, τ ∈ Σ ·,s , every (b i ) i∈|s| ∈ B s , where s = ar(σ), and every ∈ A s , then σ = τ and (b i ) i∈|s| = (c j ) j∈|t| (i.e., "outside of A" the second axiom of the notion of partial Dedekind-Peano is satisfied). Remark 35. The condition FC1 in Proposition 36 entails that the weak subalgebra A of B is even a relative subalgebra of B. Relative subalgebras satisfying the condition FC1 will be called normal according to Schmidt [17].
In connection with the functor F Σ from PAlg(Σ) to Alg(Σ) we have the following generalized recursion theorem, which we will also call the Schmidt construction. This Before defining the notion of Σ-completion that will be used in this paper, it is worth recalling that in [6], on p. 216, Burmeister says ". . . it cannot be denied that the topic of completions of partial algebras or questions about their weak, full or even normal embeddability into a total algebras of some prespecified kind are very important, in particular in computer science, but also in mathematics in general", and in [6], on p. 217, Burmeister says that a Σ-algebra B is a weak completion of a partial Σ-algebra A, if A is a weak subalgebra of B, a completion, if A is a relative subalgebra of B, and a normal completion, if A is a normal in B (for the notion of normal subalgebra see Remark 35). If A generates its (weak) completion B, then Burmeisteir says that B is a minimal (weak or normal) completion of A.
C is a wide subcategory of PAlg(Σ), i.e., a subcategory of PAlg(Σ) such that Ob(C) = Ob(PAlg(Σ)), satisfying that 1.1 for every homomorphism f of C if f = h • g, for some (g, h) ∈ E × M, then h is a homomorphism of C.

2.
F is a functor from C to Alg(Σ).

3.
η is a natural transformation from In C to In Alg(Σ) • F, i.e., for every homomorphism f : A / / B between partial Σ-algebras of C, we have that Although in the definition of the notion of Σ-completion the first coordinate is univocally determined by the second, since it is the domain of the latter, and is therefore redundant, we prefer, for the sake of clarity in the formal definition, to keep it.

Remark 38.
The conditions set forth in the third item of Definition 38 state, using the terminology of Eilenberg& Mac Lane [9], that In C is a subfunctor of the functor In Alg(Σ) • F. Regarding the use of the term "completion", it would be convenient to recall that the explanation of why Z, the profinite completion of Z, is a completion is that Z is equipped with a topology such that Z is a dense subgroup of Z.
where id L : L =⇒ L is the identity natural transformation at L, • denotes both the composition of functors and the vertical composition of natural transformations and * denotes the horizontal composition of natural transformations.
Proof. Let (D, G, ρ) be a Σ-completion. We will prove that there exists a unique morphism from (PAlg(Σ), F Σ , η) to (D, G, ρ) by gradually constructing it. By Definition 38, D is a subcategory of PAlg(Σ), therefore we have the inclusion functor In D from D to PAlg(Σ).
Moreover, we have the identity natural transformation of the functor In D , i.e., id In D , which is a natural transformation from In D to Id PAlg(Σ) • In D , this last composition being equal to In D . And, for every many-sorted partial Σ-algebra A, id In D is such that (id In D ) A , which is equal to id A , is a dense injective endomorphism of A.
We next define a natural transformation α from F Σ • In D to G. Let A be a partial Σ-algebra. Consider G(A), the completion of A, according to the Σ-completion (D, G, ρ). Then, by the universal property of the free completion, there exists a unique homomorphism (ρ A ) fc : such that the following diagram commutes We let α A stand for ρ A fc . Then to show that α = (α A ) A∈D is a natural transformation we need to check that, for every homomorphism f : A / / B of D, the following diagram in since, by Proposition 34, F Σ (A) = Sg T Σ (A) (A), it suffices to check that However, the following chain of equalities holds In the just stated chain of equalities the first equality follows from the associativity of the composition of S-sorted mappings; the second equality follows by Corollary 5; the third equality follows from the associativity of the composition of S-sorted mappings; the fourth equality follows by Proposition 34; the fifth equality follows by Definition 38; the sixth equality follows by Proposition 34; and, finally, the last equality follows from the associativity of the composition of S-sorted mappings. Therefore ( †) holds. Thus α is a natural transformation from F Σ • In D to G. Finally, for every partial Σ-algebra A, the following chain of equalities holds . This morphism is depicted in the diagram of Figure 13.
Moreover, if a Σ-completion (D, G, ρ) is such that, for every Σ-algebra A of D, G(A) is a minimal normal completion of Im(ρ A ) ∼ = A or, what is equivalente, by a result of Burmeister contained in [6] on p. 220, Ker((ρ A ) fc ) ⊆ Ker((in A,A • ) fc ), and Im((ρ A ) fc ) = G(A), then there exists a unique homomorphism coast G(A) ((in A,A • ) fc ), the coastriction of (in A,A • ) fc to G(A), from G(A) to A • such that the following diagram commutes

The Schmidt homomorphism associated to a pair formed by a morphism of Σ-completions and a homomorphism
In this section we begin to realize our project of showing the functionality of the Schmidt construction. To do it, we begin by associating to an ordered pair (α, f ), where α = (K, γ, α) is a morphism of Σ-completions from F = (C, F, η) to G = (D, G, ρ) and f a homomorphism of D from the partial Σ-algebra A to the partial Σ-algebra B, a homomorphism Υ G,0  Figure 14 commutes, where, with the customary abuse of notation, the same symbol is used for the homomorphism η K(A) • γ A from A to F(K(A)) and its corestriction to We will call Υ G,0 α ( f ) : Sch α ( f ) / / B the initial extension of f with respect to α and G or the Schmidt homomorphism of f with respect to α and G and Sch α ( f ) the Schmidt partial Σ-algebra associated to f with respect to α.
Proof. Let us recall that, since α is a natural transformation from F • K to G, for the homomorphism f : A / / B of D, the diagram in Figure 15 commutes. We let Υ G,0 We next prove that the image of the composition η K(A) • γ A is included in Sch α ( f ). In order to obtain this result we will prove the following equality Let us note that the following chain of equalities holds Figure 14. The Schmidt homomorphism of f with respect to α and G (and relatives). Figure 15. The natural transformation α at f .
In the just stated chain of equalities the first equality follows from the fact that α is a morphism from F to G, thus, for every many-sorted partial Σ-algebra A, it holds that α A • η K(A) • γ A = ρ A ; finally, the last equality follows from the fact that ρ is a natural transformation from In D to In Alg(Σ) • G.
We conclude that the image of η K(A) • γ A is included in Sch (K,γ,α) ( f ).
According to the definition of Υ G,0 α ( f ), the next equality follows This completes the proof.
Our next aim is to prove that there exists an endofunctor, denoted Υ G,0 α , of Mor tw (D), the twisted morphism category of D, thus showing the naturalness of the previous construction. For this purpose we first recall the definition of Mor tw (D).
Definition 42. Let D be a category. Then we denote by Mor tw (D) the category defined as follows 1.

2.
Morphisms from the object f : A / / B to the object f : A / / B : the ordered triples and h a homomorphism of D from B to B such that f = h • f • g, i.e., such that the diagram in Figure 16 commutes. a morphism from f to f . Then the following equation holds Proof. The following chain of equalities holds In the just stated chain of equalities, the first equality follows by associativity of the composition of S-sorted mappings; the second equality follows from the fact that F is a Σ-completion, therefore F(K(g)) • η K(A) = η K(A ) • K(g); the third equality follows by associativity of the composition of S-sorted mappings; the fourth equality follows from the fact that α is a morphism from F to G therefore, since γ is a natural transformation from In D to In C • K, it holds that K(g) • γ A = γ A • g; finally, the last equality follows by associativity of the composition of S-sorted mappings. This completes the proof of Proposition 43.
Remark 40. We suggest the reader to retain the equality presented in Proposition 43 because it will be used to justify certain equalities in the proof of the following proposition. In addition, the reader will find all the homomorphisms involved in the just mentioned equality, as well as their interrelationships with other homomorphisms, in Figure 17.
We next prove that the construction associated to a morphism of Σ-completions presented in Proposition 42 is in fact functorial. But before proceeding to do so, we establish the following auxiliary definition.
Definition 43. Let α = (K, γ, α) be a morphism of Σ-completions from F = (C, F, η) to G = (D, G, ρ). Then we denote by Υ G,0 α the mapping that sends 1. an which is an object of the same category, and 2.
a morphism (g, h) of Mor tw (D) from the object f to the object f to the morphism (F(K(g)) • in Sch α ( f ) , h) from the object Υ G,0 α ( f ) to the object Υ G,0 α ( f ), which is a morphism of the same category.
The transformations associated to the morphism of Σ-completions α are represented in Figure  18.  A A In fact, by definition, we have that Hence, to prove that the image of But, since, by Proposition 42, the many-sorted partial Σ-algebra ]generated and G( f ) • α A , F(K(g)) and in Sch α ( f ) are homomorphisms, it suffices to prove that, for every sort s ∈ S and every a ∈ A s , the following statement holds But we have that the following chain of equalities holds In the just stated chain of equalities, the first equality follows from the fact that, by Proposition 42, in Sch α ( f ) • η K(A) • γ A = η K(A) • γ A ; the second equality follows from the fact that, by Proposition 43, F(K(g)) • η K(A) • γ A = η K(A ) • γ A • g; finally, the last equality follows from the fact that, by Proposition 42, we have that Therefore, to prove (Cl1) is equivalent to prove that, for every sort s ∈ S and every a ∈ A s , the following statement holds ρ B s f s (g s (a)) ∈ ρ B s B s .
But, for every sort s ∈ S and every a ∈ A s , we have that f s (g s (a)) ∈ B s . From which it follows that ρ B s ( f s (g s (a))) ∈ ρ B s [B s ]. Thus we can affirm that the image of the homomorphism F(K(g)) • in Sch α ( f ) from Sch α ( f ) to F(K(A )) is included in Sch α ( f ). To shorten notation, we let F(K(g)) • in Sch α ( f ) stand for cores Sch α ( f ) (F(K(g)) • in Sch α ( f ) ), the corestriction of F(K(g)) • in Sch α ( f ) to Sch α ( f ). This homomorphism is represented in the diagram of Figure 17 as an unnamed dashed arrow.
In fact, since, by Proposition 42, the many-sorted partial Σ-algebra ]generated and h, Υ G,0 α ( f ), F(K(g)) and in Sch α ( f ) are homomorphisms, to prove Claim 2 it suffices to prove that Starting from the right hand side of the above equation, we have that the following chain of equalities holds In the just stated chain of equalities, the first equality follows from the fact that, by Proposi- the second equality follows from the fact that, by Proposition 43, the third equality follows from the fact that, by Proposition 42, Υ G,0 ; the fourth equality follows from the fact that (g, h) is a morphism of Mor tw (D) from f to f ; finally, the last equality follows from the fact that, by Proposition 42, Υ G,0 Therefore Υ G,0 α maps objects to objects and morphisms to morphisms. We next prove that Υ G,0 α preserves identities.
Claim 3. Let f : A / / B be a homomorphism between partial Σ-algebras of D. Then We recall that id f , the identity at f in Mor tw (D), is (id A , id B ). Let us note, that, on the one hand, we have that and, on the other hand, we have that Hence, to check that Υ G,0 But, by Proposition 42, the many-sorted partial Σ-algebra ]generated and F(K(id A )), in Sch α ( f ) , and in Sch α ( f ) are homomorphisms. Therefore, it suffices to prove that  Figure 19. The functor associated to the morphism α on compositions.
However, the following chain of equalities holds In the just stated chain of equalities, the first equality follows from the fact that, by Proposi- the second equality follows from the fact that, by Proposition 43, the third equality follows from the fact that id A is an identity; finally the last equality follows from the fact that, by it suffices to prove that But, since, by Proposition 42, the many-sorted partial Σ-algebra ]generated and F(K(g • g)), in Sch α ( f ) , F(K(g )), in Sch α ( f ) and F(K(g)) are homomorphisms, it suffices to prove that Now, starting from the right hand side of the above equation, we have that the following chain of equalities holds In the just stated chain of equalities, the first equality follows from the fact that, by Proposi- the second equality follows from the fact that, by Proposition 43, F(K(g)) • η K(A) • γ A = η K(A ) • γ A • g; the third equality follows from the fact that, by Proposition 42, in Sch α ( f ) • η K(A ) • γ A = η K(A ) • γ A ; the fourth equality follows from the fact that, by Proposition 43, F(K(g )) the fifth equality follows from the fact that, by Proposition 43, finally, the last equality follows from the fact that, by Proposition 42, Claim 4 follows. This completes the proof of Proposition 44.

Functoriality of the Schmidt construction
In this section we finally reach the objective expressed in the title of the paper. Specifically, we will prove that, for every Σ-completion G, there exists a functor Υ G from the comma category (Cmpl(Σ)↓G) to End(Mor tw (D)), the category of endofunctors of Mor tw (D), such that Υ G,0 , the object mapping of Υ G , sends an object α : F / / G of (Cmpl(Σ)↓G), i.e., a morphism of Σ-completion of Cmpl(Σ) with codomain G, to the Schmidt endofunctor Υ G,0 α in End(Mor tw (D)). To obtain the result just mentioned we first recall the definition of the comma category (Cmpl(Σ)↓G).
Definition 44. Let G be a Σ-completion of Cmpl(Σ). Then we let (Cmpl(Σ)↓G) denote the comma category of objects over G. Thus (Cmpl(Σ)↓G) is the category whose objects are morphisms α of Cmpl(Σ) whose codomain is G and whose morphisms from the object α 0 : F 0 / / G to the object α 1 : F 1 / / G are the ordered triples (α 0 , β, α 1 ), which to shorten notation we identify with β, where β is a morphism from F 0 to F 1 such that   i.e., such that the diagram in Figure 20 commutes. Let us point out that, as a morphism of (Cmpl(Σ)↓G), the domain of β is α 0 and its codomain is α 1 .
We will next show how to associate to a morphism β of (Cmpl(Σ)↓G) from an object α 0 of (Cmpl(Σ)↓G) to another object α 1 of (Cmpl(Σ)↓G) a natural transformation from Υ G,0 α 0 , the Schmidt functor at Mor tw (D) associated to its domain, to Υ G,0 α 1 , the Schmidt functor at Mor tw (D) associated to its codomain.
The reader is advised to consult the diagrams in Figures 21 and 22 for a better understanding of the entities under consideration. On the other hand, from α 1 β = α 0 , we conclude that Sch α 0 ( f ) K 1 (A) Figure 25. The natural transformation β at K 1 ( f ).
To begin with, let us consider the value of the natural transformation β, from F 0 • L to F 1 , at K 1 (A), i.e., the homomorphism β K 1 (A) from F 0 (L(K 1 (A))) to F 1 (K 1 (A)). This homomorphism is such that and we depict it in the diagram of Figure 25.
In fact, by Proposition 42, we have that Therefore, to prove that, for every s ∈ S and every z ∈ Sch α 0 ( f ) s β Let us note that the following chain of equalities holds In the just stated chain of equalities, the first equality recovers the A-th component of the natural transformation α 1 • (β * id K 1 ); finally, the last equality follows according to ( † 3 ).
In the just stated chain of equalities, the first equality applies the identity mapping at B; the second equality follows from ( † 2 ); the third equality recovers the A-th component of the natural transformation (δ * id K 1 ) • γ 1 ; the fourth equality follows from ( † 1 ); the fifth equality follows from the fact that β is a morphism from the Σ-completion F 0 to the Σ-completion F 1 ; the sixth equality follows from Proposition 42 at Υ G,0 α 1 ( f ); finally, the last equality follows from Proposition 42 at Υ G,0 α 0 ( f ). This completes the proof of Claim 6. We can now prove that Υ G,1 β is the desired natural transformation.
To prove Claim 7, we need to show that, for every pair of homomorphisms f : A / / B and f : A / / B of D and every morphism (g, h) from f to f of Mor tw (D), the following equality holds Υ G,1 β, f • Υ G,0 α 0 (g, h) = Υ G,0 α 1 (g, h) • Υ G,1 β, f .
But, taking into account the definitions of the four components of the above equality, to show that it is fulfilled is equivalent to showing that the following equalities are satisfied The reader is advised to consult the diagram in Figure 27 for a better understanding of the elements under consideration. The equation id B • h = h • id B trivially holds. Therefore, it suffices to prove the first equation. In this regard, since, by Proposition 42, the partial Σ-algebra Sch α 0 ( f ) is ]-generated and since F 1 (K 1 (g)), in Sch α 1 ( f ) , β K 1 (A) , β K 1 (A ) , F 0 (K 0 (g)), and in Sch α 0 ( f ) are homomorphisms, it suffices to prove that Starting from the left hand side of the above equation, we have that the following chain of equalities holds Sch α 0 ( f ) Sch α 1 ( f ) Figure 27. The natural transformation Υ G,1 β at a morphism (g, h) from f to f of Mor tw (D).
= β K 1 (A ) • F 0 (L(K 1 (g))) • η L(K 1 (A)) 0 In the just stated chain of equalities, the first equality follows from ( † 2 ); the second equality recovers the A-th component at the natural transformation (δ * id K 1 ) • γ 1 ; the third equality follows from ( † 1 ); the fourth equality follows from the fact that β is a morphism from the Σ-completion F 0 to the Σ-completion F 1 ; the fifth equality follows from Proposition 42; the sixth equality follows from the fact that η 1 is a natural transformation from In C 1 to In Alg(Σ) • F 1 ; the seventh equality follows from the fact that β is a morphism from the Σ-completion F 0 to the Σ-completion F 1 ; the eighth equality follows from the fact that δ is a natural transformation from In C 1 to In C 0 • L; the ninth equality follows from the fact that η 0 is a natural transformation from In C 0 to In Alg(Σ) • F 0 ; the tenth equality follows from ( † 1 ); the eleventh equality recovers the A-th component at the natural transformation (δ * id K 1 ) • γ 1 ; the twelfth equality follows from ( † 2 ); and finally, the last equality follows from Proposition 42. This concludes the proof of Claim 7. this is not the case and that we are facing a situation similar to the one that occurs in group theory, in which, e.g., the centre of a group does not determine a subfunctor of the identity functor at the category of groups (it is true that if one chooses to restrict Grp, the category of groups, by using as morphisms the surjective homomorphisms, one obtains a functor, but, for the case at hand, such a solution does not seem likely to work). In this case, the most we can say is that the objects G of the category Cmpl(Σ) play the role of parameters for the functors Υ G , while the morphisms of Cmpl(Σ) are used to act as morphisms of (Cmpl(Σ)↓G), the comma categories which are the domains of such functors.