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

38as the unique functor which satisfiesAU A A A = A UId A A = A U = A AandAUλ AA A A = A Uλ A Id A A = A Uλ A = A µ A U = A µ AA .Clearly A A A = Id A A is the identity element for the category ( A A ← A A).Corollary 3.46. Let A = (A, m A , u A ) be a monad on a category A such that theunderlying functor A preserves coequalizers. Then we havefor every F ∈ Ob (( A A ← A A)).AA A ◦ F = F and F ◦ A A A = FProof. By Proposition 3.45 we have that A A A = Id A A is the identity element for thecategory ( A A ← A A). Therefore, in particular, we have thatfor every F ∈ Ob (( A A ← A A)).AA A ◦ F = F and F ◦ A A A = FProposition 3.47. Let A = (A, m A , u A ) be a monad on a category A such that theunderlying functor A preserves coequalizers, let A P A , A Q A , A T A ∈ Ob (( A A ← A A))and let A f A : A P A → A Q A , A g A : A Q A → A T A be morphisms in ( A A ← A A). ThenAg A ◦ A f A is still a morphism in the category ( A A ← A A) and(20) A (g ◦ f) A = A g A ◦ A f A .Proof. We will prove that A g A ◦ A f A = A (g ◦ f) A where g ◦ f is an A-bilinear functorialmorphism as composite of A-bilinear functorial morphisms. By assumption,using notations of Proposition 3.42 we have the following serially commutative diagramP A A UfA A UQA A UgA A UT A A Uµ A P AUP A Uλ A µ A QAUQ A Uλ Aµ A T AUThen A f A is the unique morphism such thatwhereP A U p Pf A UQ A U pQ g A UP Af AQ Ag A T A U p TT AT A Uλ AAU A f A = f A(21) f A ◦ p P = p Q ◦ (f A U)and A g A is the unique morphism such thatwhereAU A g A = g A(22) g A ◦ p Q = p T ◦ (g A U) .□□