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

220• 2-cells: monad functor transformations in C.Remark 10.5. We denote by (X, 1 X ) in C the trivial monad on the object X withtrivial multiplication and unit m 1X : 1 1X · 1 1X → 1 1X and u 1X : 1 X → 1 1X .Definition 10.6. Let C, C ′ be two 2-categories and let G : C → C ′ be a pseudofunctor.Then the pseudofunctor Mnd (G) : Mnd (C) → Mnd (C ′ ) is defined asfollows:• Mnd (G) (X, A) = (G (X) , G (A))• Mnd (G) (Q, φ) = (G (Q) , G (φ))• Mnd (G) (σ) = G (σ) .Remark 10.7. Note that Cmd (C) = Mnd (C ∗ ) where C ∗ is the dual reversing2-cells of C.1<strong>1.</strong> Construction of BIM (C)The idea of defining this bicategory goes back to the strict monoidal category ofbalanced bimodule functors that we defined in Subsection 3.<strong>2.</strong> We observed that,considering bimodule functors with respect to the same monad on both sides, wehave a unit and a composition, so that they form a strict monoidal category. In thecase we consider a bimodule with respect to two different monads, the unit objectfails and the composition between them is no longer inside the class of objects of thecategory. The way to solve this problem is to look at balanced bimodule functorsas 0-cells of a bicategory, changing the definition of their product.Definition 1<strong>1.</strong><strong>1.</strong> Let X be a 0-cell and let (Y, B) be a monad in C. A left B-modulein C (or simply a left B-module) is a pair (Q, λ Q ) where Q : X → Y is a 1-cell andλ Q : B · Q → Q is a 2-cell in C, satisfying the associativity and unitality propertieswith respect to the monad B, i.e.λ Q (m B · 1 Q ) = λ Q (1 B · λ Q ) and λ Q (u B · 1 Q ) = 1 Q .Definition 1<strong>1.</strong><strong>2.</strong> Let (X, A) and (Y, 1 Y ) be monads in C. A right A-module in C(or simply a right A-module) is a monad functor in C ∗ , i.e. a 1-cell Q : X → Yand a 2-cell ρ Q : Q · A → 1 Y · Q = Q in C, satisfying the associativity and unitalityproperties with respect to the monad A, i.e.ρ Q (1 Q · m A ) = ρ Q (ρ Q · 1 A ) and ρ Q (1 Q · u A ) = 1 Q .Definition 1<strong>1.</strong>3. Let (X, A) and (Y, B) be monads in C. A B-A-bimodule in C (orsimply a B-A-bimodule) is a triple (Q, λ Q , ρ Q ) where• (Q, λ Q ) is a left B-module in C• (Q, ρ Q ) is a right A-module in C• the compatibility condition holdsλ Q (1 B · ρ Q ) = ρ Q (λ Q · 1 A ) .Lemma 1<strong>1.</strong>4. Let (X, A) , (Y, B) be monads in C and let (Q, λ Q ) be a left A-moduleand (Q, ρ Q ) be a right B-module. Then (Q, λ Q ) = Coequ C (m A · 1 Q , 1 A · λ Q ) and(Q, ρ Q ) = Coequ C (1 Q · m B , ρ Q · 1 B ).