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

16and[m A ◦ (u A A)] (M) = (M ⊗ R m A ) ◦ (M ⊗ R A ⊗ R u A ) ◦ r −1M⊗ R A= (M ⊗ R [m A ◦ (A ⊗ R u A )]) ◦ r −1M⊗ R A= (M ⊗ R r A ) ◦ r −1M⊗ R A = M ⊗ R A = AM.Proposition 3.4 ([H]). Let (L, R) be an adjunction with unit η and counit ɛ whereL : B → A and R : A → B. Then RL = (RL, RɛL, η) is a monad on the category B.Proof. We have to prove that(RɛL)◦(RLRɛL) = (RɛL)◦(RɛLRL) and (RɛL)◦RLη = RL = (RɛL)◦(ηRL) .In fact we haveand(RɛL) ◦ (RLRɛL) ɛ = (RɛL) ◦ (RɛLRL)(RɛL) ◦ RLη (L,R)= RL (L,R)= (RɛL) ◦ (ηRL) .Definition 3.5. A left module functor for a monad A = (A, m A , u A ) on a categoryA is a pair ( Q, A µ Q)where Q : B → A is a functor and A µ Q : AQ → Q is a functorialmorphism satisfying:A µ Q ◦ ( A A µ Q)= A µ Q ◦ (m A Q) and Q = A µ Q ◦ (u A Q) .Definition 3.6. A right module functor for a monad A = (A, m A , u A ) on a categoryA is a pair ( P, µ A P)where P : A → B, is a functor and µAP : P A → P is a functorialmorphism such thatµ A P ◦ ( µ A P A ) = µ A P ◦ (P m A ) and P = µ A P ◦ (P u A ) .Remark 3.7. Let A = (A, m A , u A ) be a monad on a category A and let ( )Q, A µ Q bea left A-module functor and ( )P, µ A P be a right A-module functor. By the unitalityproperty of A µ Q and µ A P we deduce that they are both epimorphism.Definition 3.8. For two monads A = (A, m A , u A ) on a category A and B =(B, m B , u B ) on a category B, a A-B-bimodule functor is a triple ( Q, A µ Q , µ Q) B , whereQ : B → A is a functor and ( ) ( )Q, A µ Q is a left A-module functor, Q, µBQ is a rightB-module functor such that in additionA µ Q ◦ ( )Aµ B Q = µBQ ◦ (A µ Q B ) .Definition 3.9. A module for a monad A = (A, m A , u A ) on a category A is a pair(X, A µ X)where X ∈ A and A µ X : AX → X is a morphism in A such thatA µ X ◦ ( A A µ X)= A µ X ◦ (m A X) and X = A µ X ◦ (u A X) .A morphism between two A-modules ( X, A µ X)and(X ′ , A µ X ′)is a morphism f :X → X ′ in A such thatA µ X ′ ◦ (Af) = f ◦ A µ X .We will denote by A A the category of A-modules and their morphisms. This is theso-called Eilenberg-Moore category which is sometimes also denoted by A A .□