Abstract
After proving, in a purely categorial way, that the inclusion functor |$\textrm {In}_{\textbf {Alg}(\varSigma )}$| from |$\textbf {Alg}(\varSigma )$|, the category of many-sorted |$\varSigma $|-algebras, to |$\textbf {PAlg}(\varSigma )$|, the category of many-sorted partial |$\varSigma $|-algebras, has a left adjoint |$\textbf {F}_{\varSigma }$|, the (absolutely) free completion functor, we recall, in connection with the functor |$\textbf {F}_{\varSigma }$|, the generalized recursion theorem of Schmidt, which we will also call the Schmidt construction. Next, we define a category |$\textbf {Cmpl}(\varSigma )$|, of |$\varSigma $|-completions, and prove that |$\textbf {F}_{\varSigma }$|, labelled with its domain category and the unit of the adjunction of which it is a part, is a weakly initial object in it. Following this, we associate to an ordered pair |$(\boldsymbol {\alpha },f)$|, where |$\boldsymbol {\alpha }=(K,\gamma,\alpha )$| is a morphism of |$\varSigma $|-completions from |${{\mathscr {F}}}=(\textbf {C},F,\eta )$| to |$\mathscr {G}= (\textbf {D},G,\rho )$| and |$f$| a homomorphism of |$\textbf {D}$| from the partial |$\varSigma $|-algebra |$\textbf {A}$| to the partial |$\varSigma $|-algebra |$\textbf {B}$|, a homomorphism |$\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}(f)\colon \textbf {Sch}_{\boldsymbol {\alpha }}(f)\longrightarrow \textbf {B}$|. We then prove that there exists an endofunctor, |$\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}$|, of |$\textbf {Mor}_{\textrm {tw}}(\textbf {D})$|, the twisted morphism category of |$\textbf {D}$|, thus showing the naturalness of the previous construction. Afterwards, we prove that, for every |$\varSigma $|-completion |$\mathscr {G}=(\textbf {D},G,\rho )$|, there exists a functor |$\varUpsilon ^{\mathscr {G}}$| from the comma category |$(\textbf {Cmpl}(\varSigma )\!\downarrow \!\mathscr {G})$| to |$\textbf {End}(\textbf {Mor}_{\textrm {tw}}(\textbf {D}))$|, the category of endofunctors of |$\textbf {Mor}_{\textrm {tw}}(\textbf {D})$|, such that |$\varUpsilon ^{\mathscr {G},0}$|, the object mapping of |$\varUpsilon ^{\mathscr {G}}$|, sends a morphism of |$\varSigma $|-completion of |$\textbf {Cmpl}(\varSigma )$| with codomain |$\mathscr {G}$|, to the endofunctor |$\varUpsilon ^{\mathscr {G},0}_{\boldsymbol {\alpha }}$|.