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

Since we are assuming that the coequalizers are preserved by the composition withany 1-cell, both the rows are coequalizers and the left square serially commutes. Infact(λ Q · 1 Q ′) (1 A · ρ Q · 1 Q ′) Qbim= (ρ Q · 1 Q ′) (λ Q · 1 B · 1 Q ′)(λ Q · 1 Q ′) (1 A · 1 Q · λ Q ′) λ Q= (1 Q · λ Q ′) (λ Q · 1 B · 1 Q ′) .By the universal property of the coequalizer(Q • B Q ′ , p Q,Q ′) = Coequ C (ρ Q · 1 Q ′, 1 Q · λ Q ′), there exists a unique 2-cell λ Q•B Q ′ :A · Q • B Q ′ → Q • B Q ′ such thatλ Q•B Q ′ (1 A · p Q,Q ′) = p Q,Q ′ (λ Q · 1 Q ′) .By similar computations, one can prove that (Q • B Q ′ , λ Q•B Q ′) is a left A-module.Finally, we prove that the structures are compatible. In factρ Q•B Q ′ (λ Q• B Q ′ · 1 C) (1 A · p Q,Q ′ · 1 C )(236)= ρ Q•B Q ′ (p Q,Q ′ · 1 C) (λ Q · 1 Q ′ · 1 C )(237)= p Q,Q ′ (1 Q · ρ Q ′) (λ Q · 1 Q ′ · 1 C ) λ Q= p Q,Q ′ (λ Q · 1 Q ′) (1 A · 1 Q · ρ Q ′)(236)= λ Q•B Q ′ (1 A · p Q,Q ′) (1 A · 1 Q · ρ Q ′)223(237)= λ Q•B Q ′ (1 A · ρ Q•B Q ′) (1 A · p Q,Q ′ · 1 C )and since 1 A · p Q,Q ′ · 1 C is epi, we get thatρ Q•B Q ′ (λ Q• B Q ′ · 1 C) = λ Q•B Q ′ (1 A · ρ Q•B Q ′)i.e. (Q • B Q ′ , λ Q•B Q ′, ρ Q• B Q ′) is an A-C-bimodule.□Proposition 1<strong>1.</strong>6. Let (X, A) and (Y, B) be monads in the 2-category C, let (Q, λ Q )be a left A-module and (Q, ρ Q ) be a right B-module. Then A• A Q ≃ Q and Q• B B ≃Q.Proof. Let us consider the trivial left A-module (A, m A ) and note that (A • A Q, p A,Q ) =Coequ C (m A · 1 Q , 1 A · λ Q ). We already observed, in Lemma 1<strong>1.</strong>4, that (Q, λ Q ) =Coequ C (m A · 1 Q , 1 A · λ Q ). Therefore, there exists an isomorphism l Q : A • A Q → Qsuch that(238) l Q p A,Q = λ Q .Similarly, if we consider the trivial right B-module (B, m B ), since (Q • B B, p Q,B ) =Coequ C (1 Q · m B , ρ Q · 1 B ) = (Q, ρ Q ) we deduce that there exists an isomorphismr Q : Q • B B → Q such that(239) r Q p Q,B = ρ Q .Proposition 1<strong>1.</strong>7. Let (X, A), (Y, B), (Z, C), (W, D) be monads in the 2-categoryC and let (Q, λ Q , ρ Q ) be an A-B-bimodule, (Q ′ , λ Q ′, ρ Q ′) be a B-C-bimodule and(Q ′′ , λ Q ′′, ρ Q ′′) be a C-D-bimodule. Then the coequalizers (Q • B Q ′ ) • C Q ′′ ≃ Q • B(Q ′ • C Q ′′ ) are isomorphic as A-D-bimodules.□