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

95We have( A Uλ A ) ◦ ( Aσ A A)◦ (AQpP ) defσA A= ( A Uλ A ) ◦ (A A Uλ A ) ◦ ( Aσ A AAU )AUλ A coequ= ( A Uλ A ) ◦ (m AA U) ◦ ( Aσ A AU ) (80)= ( A Uλ A ) ◦ ( σ A AU ) ◦ (A µ Q P A U )defσA= A σA A ◦ (Qp P ) ◦ (A µ Q P A U ) A µ Q= σAA ◦ (A )µ Q P A ◦ (AQpP )and since A, Q preserve coequalizers, AQp P is an epimorphism, so that we get( A Uλ A ) ◦ ( Aσ A A)= σAA ◦ (A µ Q P A).Hence, by Lemma 3.29, there exists a unique morphism A σA A: AQP A → Id A A suchthatAU A σA A = σA.ANow, note that, by definition of σA A , we haveσA A ◦ (Qp P ) = ( A Uλ A ) ◦ ( σ A AU )so that by applying it to A F we get(σAA AF ) ◦ (Qp P A F ) = ( A Uλ AA F ) ◦ ( σ A AU A F ) .Hence, by Proposition 3.34, we obtain that(σAA AF ) ◦ ( Qµ A P)= mA ◦ ( σ A A ) (80)= σ A ◦ ( Qµ A P).Since Qµ A P is an epimorphism, we deduce that σA AAF = σ A .Proposition 6.16. Let A and B be categories with equalizers and let τ : Q → QP Qbe a regular herd for a formal dual structure M = (A, B, P, Q, σ A , σ B ) where theunderlying functors P : A → B, Q : B → A and A : A → A preserve equalizers.Let• C = ( C, ∆ C , ε C) be the comonad on the category A constructed in Proposition6.1;• ( Q, C ρ Q)be the left C-comodule functor constructed in Proposition 6.1;• A Q : B → A A be the functor defined in Lemma 3.29;• Φ : AC → CA be the mixed distributive law between the comonad C and themonad A constructed in Proposition 6.14;• ˜C be the lifting of C on the category A A constructed in Theorem 5.7.Then there exists a functorial morphism eC ρ A Q : A Q → ˜C A Q such thatAU eC ρ A Q = C ρ Q .( )Moreover, AQ, eC ρ A Q is a left ˜C-comodule functor.Proof. Since τ : Q → QP Q is a regular herd for M = (A, B, P, Q, σ A , σ B ), byProposition 6.14, the mixed distributive law Φ : AC → CA is uniquely defined by(iA) ◦ Φ = ( QP σ A) ◦ (τP ) ◦ (A µ Q P ) ◦ (Ai) .Now we prove that C ρ Q yields a functorial morphism eC ρ A Q. In fact we have(iQ) ◦ ( C A µ Q)◦ (ΦQ) ◦(A C ρ Q) i= ( QP A µ Q)◦ (iAQ) ◦ (ΦQ) ◦(A C ρ Q)□