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

36satisfyingf B ◦ p P = p Q ◦ (f B U) .Then we can considerAf B : A P B → A Q Bsuch thatAU A f B = f B .Proof. Consider the following diagramP B B Uµ B P BUP B Uλ B P B U p PP Bf BfB B UQB B Uµ B QBUf B UQ B Uλ B Q B U pQ Q BSince f is a functorial morphism and it is a functorial morphism of right B-modulefunctors, the left square serially commutes. Note thatp Q ◦ (f B U) ◦ ( µ B P BU ) = p Q ◦ (f B U) ◦ (P B Uλ B )so that, by the universal property of the coequalizer, there exists a unique morphismf B : P B → Q B such that(18) f B ◦ p P = p Q ◦ (f B U) .We now want to prove that f B is a functorial morphism of left A-module functor.In fact we havef B ◦ A µ PB ◦ (Ap P ) (7)= f B ◦ p P ◦ (A µ P B U )(18)= p Q ◦ (f B U) ◦ (A µ P B U ) fleftAlin= p Q ◦ (A µ QB U ) ◦ (Af B U)(7)= A µ QB ◦ (Ap Q ) ◦ (Af B U) (18)= A µ QB ◦ (Af B ) ◦ (Ap P )and since A preserves coequalizers Ap Q is an epimorphism so that we getf B ◦ A µ PB = A µ QB ◦ (Af B ) .Then there exists a functorial morphism A f B : A P B → A Q B such thatAU A f B = f B .3.<strong>2.</strong> The category of balanced bimodule functors. We will construct here themonoidal category of balanced bimodule functors with respect to a monad.Definition 3.43. Let A = (A, m A , u A ) be a monad over a category A such thatAhas coequalizers and the underlying functor A preserves coequalizers. Let us definethe category ( A A ← A A) of balanced bimodule functors as followsOb Objects are functors A Q A : A A→ A A where Q : A → A is an A-A-bimodulefunctor such that Q A preserves coequalizers.M Morphisms are functorial morphisms A f A : A P A → A Q A where f : P → Q isa functorial morphism of A-A-bimodule functors.□