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

81following diagram0Hom B (Y, Y ′ )Hom B (Y,ηY ′ )Hom B (Y, RLY ′ )K Y,Y ′K Y,RLY ′0HomC A (K ϕ Y, K ϕ Y ′ )Hom CA (K ϕY,K ϕηY ′ )HomC A (K ϕ Y, K ϕ RLY ′ )Hom B (Y,RLηY ′ )Hom B (Y,ηRLY ′ )Hom B (Y, RLRLY ′ )Hom CA (K ϕY,K ϕRLηY ′ )K Y,RLRLY ′Hom CA (K ϕY,K ϕηRLY ′ ) HomC A (K ϕ Y, K ϕ RLRLY ′ )Since (Y ′ , ηY ′ ) = Equ B (ηRLY ′ , RLηY ′ ) the left column of the diagram is exact byLemma <strong>2.</strong>17. By Lemma 4.65 we have (K ϕ Y, K ϕ ηY ) = EquC A (K ϕ ηRLY, K ϕ RLηY )so that also the right column is also exact by Lemma <strong>2.</strong>17. Let f ∈ Hom B (Y, Y ′ )and g ∈ Hom B (Y, RLY ′ ). SinceK ϕ (ηY ′ ◦ f) = (K ϕ ηY ′ ) ◦ (K ϕ f) ,K ϕ (ηRLY ′ ◦ g) = (K ϕ ηRLY ′ ) ◦ (K ϕ g) and K ϕ (RLηY ′ ◦ g) = (K ϕ RLηY ′ ) ◦ (K ϕ g)the diagram is serially commutative. By Proposition 4.66, K Y,RLY ′ and K Y,RLRLY ′are isomorphisms and so is K Y,Y ′ by Lemma <strong>2.</strong>15. □Corollary 4.69 (Beck’s Theorem for comonads). Let (L, R) be an adjunctionwhere L : B → A and R : A → B. Then K = Υ (Id LR ) : B → LR A is full andfaithful if and only if for every Y ∈ B we have that (Y, ηY ) = Equ B (ηRLY, RLηY ) .5.<strong>1.</strong> Distributive laws.5. Liftings and distributive lawsDefinition 5.<strong>1.</strong> Let A = (A, m, u) be a monad and C = (C, ∆, ε) be a comonadon the same category A. A functorial morphism Φ : AC → CA is called a mixeddistributive law (or in some papers an entwining) if• Φ ◦ (mC) = (Cm) ◦ (ΦA) ◦ (AΦ) and Φ ◦ (uC) = Cu• (∆A) ◦ Φ = (CΦ) ◦ (ΦC) ◦ (A∆) and (εA) ◦ Φ = Aε.Definition 5.<strong>2.</strong> Let A = (A, m, u) be a monad and C = (C, ∆, ε) be a comonadon the same category A. A functorial morphism Ψ : CA → AC is called an oppositemixed distributive law if• Ψ ◦ (Cm) = (mC) ◦ (AΨ) ◦ (ΨA) and Ψ ◦ (Cu) = uC• (A∆) ◦ Ψ = (ΨC) ◦ (CΨ) ◦ (∆A) and (Aε) ◦ Ψ = εA.Lemma 5.3. Let A = (A, m A , u A ) be a monad and let C = ( C, ∆ C , ε C) be a comonadon the category A. Let Q : B → A be a functor such that ( Q, A µ Q)is left A-module functor. Assume that Φ : AC → CA is a mixed distributive law. Then(CQ, A µ CQ)=(CQ,(C A µ Q)◦ (ΦQ))is a left A-module functor.