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, labelled 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 of D from the partial Σ-algebra A to the partial Σ-algebra B⁠, a homomorphism Υ G,0α(f):Schα(f)⟶B⁠. We then prove that there exists an endofunctor, Υ G,0α, 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 ΥG 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 ΥG⁠, sends a morphism of Σ-completion of Cmpl(Σ) with codomain G⁠, to the endofunctor ΥG,0α