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

96defΦ= ( QP A µ Q)◦(QP σ A Q ) ◦ (τP Q) ◦ (A µ Q P Q ) ◦ (AiQ) ◦ ( A C ρ Q)(61),(82)= ( QP µ B Q)◦(QP QσB ) ◦ (τP Q) ◦ (A µ Q P Q ) ◦ (Aτ)τ= ( QP µ B Q)◦ (τB) ◦(QσB ) ◦ (A µ Q P Q ) ◦ (Aτ)A µ Q=(QP µBQ)◦ (τB) ◦( Aµ Q B ) ◦ ( AQσ B) ◦ (Aτ)(70)= ( QP µ B Q)◦ (τB) ◦( Aµ Q B ) ◦ (AQu B )A µ Q=(QP µBQ)◦ (τB) ◦ (QuB ) ◦ A µ Q τ = ( QP µ B Q)◦ (QP QuB ) ◦ τ ◦ A µ QQmodfun= τ ◦ A µ Q(61)= (iQ) ◦ C ρ Q ◦ A µ Qand since by construction iQ is a monomorphism we get that(C A µ Q)◦ (ΦQ) ◦(A C ρ Q)= C ρ Q ◦ A µ Q .By Lemma 5.3 we know that( )C A µ Q ◦ (ΦQ) = A µ CQso that we getA µ CQ ◦ ( A C ρ Q)=(C A µ Q)◦ (ΦQ) ◦(A C ρ Q)= C ρ Q ◦ A µ Q .Hence there exists a morphism eC ρ A Q : A Q → ˜C A Q such thatAU eC ρ A Q = C ρ Q .By the coassociativity and counitality(properties)of C ρ Q , we deduce that eC ρ A Q isalso coassociative and counital so that AQ, eC ρ A Q is a left ˜C-comodule functor. □Lemma 6.17. Let M = (A, B, P, Q, σ A , σ B ) be a formal dual structure where theunderlying functors are A : A → A, B : B → B, P : A → B and Q : B → A.Assume that both categories A and B have coequalizers and the functors A, QBpreserve them. Assume thatand• C = ( C, ∆ C , ε C) is a comonad on the category A such that C preservescoequalizers• ˜C(= ˜C, ∆C e, ε e )Cis a lifting of the comonad of C to the category A A( )• AQ, eC ρ A Q is a left ˜C-comodule functor where A U eC ρ A Q = C ρ Q .Consider the functorial morphismscan 1 := ( Cσ A) ◦ (C ρ Q P ) : QP → CA( ) ( )Acan A := ˜CA σAA eC◦ ρ A QP A : A QP A → ˜CThen can 1 is an isomorphism if and only if A can A is an isomorphism.