Contents 1. Introduction 2 2. Preliminaries 4 2.1. Some results on ...

106There exist functorial morphisms m B : BB → B and u B : B → B such thatB = (B, m B , u B ) is a monad over B that preserves coequalizers. Moreover m B andu B are uniquely determined by(108) m B ◦ (yB) = y ◦ ( )P µ B Qor equivalently(109) m B ◦ (yy) = y ◦ (P χ)and(110) y ◦ δ D = u B ◦ ε D .Moreover ( Q, µ B Q)is a right B-module functor.Proof. By left-right symmetric argument of those used in proof of Proposition 6.25,one can prove this Proposition.□Definition 6.27. Let A and B be categories. A preformal codual structure is aeightuple Θ = ( C, D, P, Q, δ C , δ D , ε C , ε D) where C : A → A, D : B → B, P : A → Band Q : B → A are functors, δ C : C → QP, δ D : D → P Q, ε C : C → A, ε D : D → Bare functorial morphisms. A copretorsor χ for Θ is a functorial morphism χ :QP Q → Q satisfying the following conditions:1) Coassociativity, in the sense that(111) χ ◦ (χP Q) = χ ◦ (QP χ)2) Counitality, in the sense that(112) χ ◦ (δ C Q) = ε C Qand(113) χ ◦ (Qδ D ) = Qε D .Definition 6.28. A preformal codual structure Θ = ( C, D, P, Q, δ C , δ D , ε C , ε D)will be called regular whenever ( A, ε C) (= Coequ Fun Cε C , ε C C ) and ( (B, ε D) =Coequ Fun Dε D , ε D D ) . In this case a copretorsor for Θ will be called a regularcopretorsor.Theorem 6.29. Let A and B be categories with coequalizers and let χ : QP Q →Q be a regular copretorsor for Θ = ( C, D, P, Q, δ C , δ D , ε C , ε D) . Assume that theunderlying functors P, Q, C and D preserve coequalizers. Let w l = (χP ) ◦ (QP δ C )and w r = QP ε C : QP C → QP . Set(114) (A, x) = Coequ Fun(w l , w r) .There exists a functorial morphism A µ Q : AQ → Q such that(115)A µ Q ◦ (xQ) = χ.There exist functorial morphisms m A : AA → A and u A : A → A such that A =(A, m A , u A ) is a monad over A that preserves coequalizers. Moreover m A and u Aare uniquely determined byx ◦ (χP ) = m A ◦ (xx) and x ◦ δ C = u A ◦ ε C .