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

123(135)= ( C ρ Q P C U) ◦ ( Qι P ) ◦ δ C CC ρ Q=(CQιP ) ◦ ( C ρ Q P C ) ◦ δ C Cand since C, Q preserve equalizers, CQι P is a monomorphism, so that we get(CδCC)◦( CUγ C) = ( C ρ Q P C ) ◦ δ C C .Hence, by Lemma 4.28, there exists a unique morphism there exists a unique morphismC δ C C : C A → C QP C such thatC U C δ C C = δ C C .Moreover, note that by definition of δC C we have( ) QιP◦ δC C = ( δ C C U ) ◦ (C Uγ C)so that by applying it to C F we get(QιP C F ) ◦ ( δ C C C F ) = ( δ C C U C F ) ◦ (C Uγ C C F ) .Hence, by Proposition 4.32, we obtain that(QρDQ)◦(δCC C F ) = (δ C C) ◦ ∆ C (125)= ( Qρ C P)◦ δC .Since Qρ D Q is a monomorphism, we deduce that δC C C F = δ C .Proposition 6.40. Let A and B be categories with coequalizers and let χ : QP Q →Q be a regular coherd for a formal codual structure X = (C, D, P, Q, δ C , δ D ) where theunderlying functors P : A → B, Q : B → A and C : A → A preserve coequalizers.Let• A = (A, m A , u A ) be the monad on the category A constructed in Proposition6.25;• ( Q, A µ Q)be the left A-module functor constructed in Proposition 6.25;• C Q : B → C A be the functor defined in Lemma 4.28;• Λ : AC → CA be the mixed distributive law between the comonad C and themonad A constructed in Proposition 6.38;• Ã be the lifting of A on the category C A constructed in Theorem 5.7.Then there exists a functorial morphism eA µC Q : ÃC Q → C Q such that□Moreover,C U eA µC Q = A µ Q .( )C Q, eA µC Q is a left Ã-module functor.Proof. Since χ : QP Q → Q is a regular coherd for X = (C, D, P, Q, δ C , δ D ), byProposition 6.38 the mixed distributive law Λ : AC → CA is uniquely defined byΛ ◦ (xC) = (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) .Now we prove that A µ Q yields a functorial morphism eA µC Q. In fact we have(C A µ Q)◦ (ΛQ) ◦(A C ρ Q)◦ (xQ) x = ( C A µ Q)◦ (ΛQ) ◦ (xCQ) ◦(QP C ρ Q)defΛ= ( C A µ Q)◦ (CxQ) ◦( Cρ Q P Q ) ◦ (χP Q) ◦ (QP δ C Q) ◦ ( QP C ρ Q)(101),(127)= (Cχ) ◦ (C ρ Q P Q ) ◦ (χP Q) ◦ (QP Qδ D ) ◦ ( QP ρ D Q)