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

116= ( Aε C Q ) ◦ ( cocan1 −1 Q ) ◦ (Qδ D ) ◦ (A µ Q D ) ◦ ( )Aρ D Q= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (A µ Q P Q ) ◦ (AQδ D ) ◦ ( )Aρ D Q(127)= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (A µ Q P Q ) ◦ (Aδ C Q) ◦ ( A C ρ Q)= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (cocan 1 Q) ◦ ( A C ρ Q)= ( Aε C Q ) ◦ ( A C ρ Q) C ρ Q counital= AQso we obtaincocan −12 = ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) .(b) ⇒ (a) Follows from Theorem 6.36. (a) ⇔ (c) follows by similar computations.□6.10. Coherds and distributive laws. The following result is a reformulation ofTheorem <strong>2.</strong>16 in [BV] in our categorical setting.Proposition 6.38. Let A and B be categories with equalizers and let χ : QP Q →Q be a regular coherd for X = (C, D, P, Q, δ C , δ D ) where the underlying functorsP : A → B, Q : B → A, C : A → A and D : B → B preserve coequalizers.Let A = (A, m A , u A ) and B = (B, m B , u B ) be the associated monads constructed inProposition 6.25 and in Proposition 6.26. Then1) There exists a mixed distributive law between the monad A and the comonadC, Λ : AC → CA such thatΛ ◦ (xC) = λ = (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) .2) There exists an opposite mixed distributive law between the monad B and thecomonad D, Γ : DB → BD such thatΓ ◦ (Dy) = γ = (yD) ◦ ( P ρ D Q)◦ (P χ) ◦ (δD P Q) .Proof. 1) Consider the functorial morphism given byQP C QP δ C−→ QP QP −→ χPQP C ρ Q P−→ CQP −→ CxCAλ = (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C )Recall that w l = (χP ) ◦ (QP δ C ) and w r = QP ε C : QP C → QP and let us provethatλ ◦ ( w l C ) ? = λ ◦ (w r C)that isLet us compute(Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) ◦ (χP C) ◦ (QP δ C C)?= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) ◦ ( QP ε C C ) .(Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP δ C ) ◦ (χP C) ◦ (QP δ C C)χ= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (χP QP ) ◦ (QP QP δ C ) ◦ (QP δ C C)δ C ,(111)= (Cx) ◦ (C ρ Q P ) ◦ (χP ) ◦ (QP χP ) ◦ (QP δ C QP ) ◦ (QP Cδ C )