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

124χ= (Cχ) ◦ (C ρ Q P Q ) ◦ (Qδ D ) ◦ (χD) ◦ ( )QP ρ D Q= (Cχ) ◦ (CQδD ) ◦ (C ρ Q D ) ◦ (χD) ◦ ( )QP ρ D QC ρ Q(130)= ( CQε D) ◦ (C ρ Q D ) ◦ (χD) ◦ ( QP ρ D QC ρ Q= C ρ Q ◦ ( Qε D) ◦ (χD) ◦ ( QP ρ D Q) χ= C ρ Q ◦ χ ◦ ( QP Qε D) ◦ ( QP ρ D QQcomfun= C ρ Q ◦ χ (101)= C ρ Q ◦ A µ Q ◦ (xQ)and since by construction xQ is an epimorphism we get that(C A µ Q)◦ (ΛQ) ◦(A C ρ Q)= C ρ Q ◦ A µ Q .By Lemma 5.5 we know that(ΛQ) ◦ ( A C ρ Q)= C ρ AQ))so that (C A µ Q)◦ C ρ AQ = ( C A µ Q)◦ (ΛQ) ◦(A C ρ Q)= C ρ Q ◦ A µ Q .Hence there exists a morphism eA µC Q : ÃC Q → C Q such thatC U eA µC Q = A µ Q .By the associativity and unitality(properties)of A µ Q , we deduce that eA µC Q is alsoassociative and unital so thatC Q, eA µC Q is a left Ã-module functor. □Lemma 6.4<strong>1.</strong> Let X = (C, D, P, Q, δ C , δ D ) be a formal codual structure with underlyingfunctors P : A → B, Q : B → A, C : A → A and D : B → B. Assume that Aand B are categories with equalizers and C, QD preserves them. Assume that• A = (A, m A , u A ) is a monad on the category A such that A preserves equalizers• ( Q, A µ Q)is a left A-module functorandis a lifting of the monad of A to the category C A( )•C Q, eA µC Q is a left Ã-module functor where C U eA µC Q = A µ Q .• Ã = (Ã, m e A, u e A)Consider the functorial morphismscocan 1 := (A µ Q P ) ◦ (Aδ C ) : AC → QP( ) (ÃC )C cocan C eA:= µC QP C ◦ δCC : Ã → C QP C .Then cocan 1 is an isomorphism if and only if C cocan C is an isomorphism.Proof. Let us consider cocan 1 := (A µ Q P ) ◦ (Aδ C ) : AC → QP. Let ( Q D , ι Q) bethe equalizer described in Proposition 4.29. Since A µ Q is a functorial morphism, wehave that (QιP ) ◦ (A µ Q P C) = (A µ Q P C U ) ◦ ( AQι P ) .Now, by Lemma 6.39, δ C induces a morphism δ C C : C U → QP C such that(QιP ) ◦ δ C C = ( δ C C U ) ◦ (C Uγ C) .