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

105We computeso that we getm A ◦ (u A A) ◦ ( ε C A ) ◦ (Cx) (103)= m A ◦ (xA) ◦ (δ C A) ◦ (Cx)δ C= m A ◦ (xA) ◦ (QP x) ◦ (δ C QP ) = m A ◦ (xx) ◦ (δ C QP )(102)= x ◦ (χP ) ◦ (δ C QP ) (99)= x ◦ (ε C QP ) εC = ( ε C A ) ◦ (Cx)m A ◦ (u A A) ◦ ( ε C A ) ◦ (Cx) = ( ε C A ) ◦ (Cx)and since ( ε C A ) ◦ (Cx) is an epimorphism we deduce thatm A ◦ (u A A) = A.Therefore we obtain that m A is unital. We computeA µ Q ◦ ( A A µ Q)◦ (AxQ) ◦ (xQP Q)(101)= A µ Q ◦ (Aχ) ◦ (xQP Q) x = A µ Q ◦ (xQ) ◦ (QP χ)(101)= χ ◦ (QP χ) (98)= χ ◦ (χP Q) (101)= A µ Q ◦ (xQ) ◦ (χP Q)(102)= A µ Q ◦ (m A Q) ◦ (xxQ) = A µ Q ◦ (m A Q) ◦ (AxQ) ◦ (xQP Q) .Since (AxQ) ◦ (xQP Q) is an epimorphism we getWe calculateA µ Q ◦ ( A A µ Q)= A µ Q ◦ (m A Q) .A µ Q ◦ (u A Q) ◦ ( ε C Q ) (103)= A µ Q ◦ (xQ) ◦ (δ C Q)(101)= χ ◦ (δ C Q) (99)= ( ε C Q ) .Since ( ε C Q ) is an epimorphism we obtainA µ Q ◦ (u A Q) = Q.Proposition 6.26. Let A and B be categories with coequalizers and let P : A → B,Q : B → A, and D : B → B be functors. Assume that all the functors P, Q and Dpreserve coequalizers. Let ε D : D → B be a functorial morphism and assume that(B, εD ) = Coequ Fun(Dε D , ε D D ) . Let χ : QP Q → Q be a functorial morphism suchthatχ ◦ (QP χ) = χ ◦ (χP Q) .Let δ D : D → P Q be a functorial morphism such that(105) χ ◦ (Qδ D ) = Qε D .Let z l = (P χ) ◦ (δ D P Q) and z r = ε D P Q : DP Q → P Q. Set(106) (B, y) = Coequ Fun(z l , z r) .There exists a functorial morphism µ B Q : QB → Q such that(107) µ B Q ◦ (Qy) = χ.□