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

40Theorem 3.49. Let A = (A, m A , u A ) be a monad over a category A such thatAhas coequalizers and the underlying functor A preserves coequalizers. The category( A A ← A A) of balanced bimodule functors is a strict monoidal category.Proof. By Proposition 3.44, we defined a composition of the objects of the category( A A ← A A). Moreover, by Proposition 3.45, A A A is the unit for the category( A A ← A A). Since the composition of functors is associative and by Corollary 3.46,it is easy to prove that ( A A ← A A) is a strict monoidal category.□3.3. The comparison functor for monads.Proposition 3.50. Let (L, R) be an adjunction where L : B → A and R : A → Bwith unit η and counit ɛ and let A = (A, m A , u A ) be a monad on the category B.There exists a bijective correspondence between the following collections of data:M monad morphisms ψ : A = (A, m A , u A ) → RL = (RL, RɛL, η)R functorial morphism r : LA → L such that (L, r) is a right module functorfor the monad AL functorial morphism l : AR → R such that (R, l) is a left module functor forthe monad Agiven byΘ : M → R where Θ (ψ) = (ɛL) ◦ (Lψ)Ξ : R → M where Ξ (r) = (Rr) ◦ (ηA)Γ : M → L where Γ (ψ) = (Rɛ) ◦ (ψR)Λ : L → M where Λ (l) = (lL) ◦ (Aη) .Theorem 3.5<strong>1.</strong> Let (L, R) be an adjunction where L : B → A and R : A → Band let A = (A, m A , u A ) be a monad on the category B. There exists a bijectivecorrespondence between the following collections of data:K Functors K : A → A B such that A U ◦ K = RM monad morphisms ψ : A = (A, m A , u A ) → RL = (RL, RɛL, η)given byΨ : K → M where Ψ (K) = ([ A Uλ A K] L) ◦ (Aη)Υ : M → K where Υ (ψ) (X) = (RX, (RɛX) ◦ (ψRX)) and Υ (ψ) (f) = Rf.Remark 3.5<strong>2.</strong> When A = RL = (RL, RɛL, η) and ψ = Id RL the functor K =Υ (ψ) : A → RL B such that RL U ◦ K = R is called the Eilenberg-Moore comparisonfunctor.Corollary 3.53. Let A = (A, m A , u A ) and B = (B, m B , u B ) be monads on acategory B. There exists a bijective correspondence between the following collectionsof data:K Functors K : A B → B B such that B U ◦ K = A U,M monad morphisms ψ : A → Bgiven byΨ : K → M where Ψ (K) = ([ A Uλ A K] A F ) ◦ (Au A )Υ : M → K where Υ (ψ) (X) = ( A UX, ( A Uλ A X) ◦ (ψ A UX)) and Υ (ψ) (f) = A U (f) .