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

138Proposition 7.7. In the setting of Theorem 6.29 and Proposition 7.6, there existtwo functorial morphisms σ A : Q ̂Q → A and σ B : ̂QQ → B where σ A is A-bilinearand σ B is B-bilinear and they fulfill(161) σ A ◦ (Ql) = m A ◦ (xA)and(162) σ B ◦ (ν ′ 0Q) = m B ◦ (By) .Moreover the associative conditions hold, that isA µ Q ◦ ( σ A Q ) = µ B Q ◦ ( Qσ B) and B µb Q◦Proof. First we want to prove thatIn fact we have( ) (σ B ̂Q = µ A Q b ◦ ̂QσA).m A ◦ (xA) ◦ (QP x) ◦ ( Qz l P ) = m A ◦ (xA) ◦ (QP x) ◦ (Qz r P ) .m A ◦ (xA) ◦ (QP x) ◦ ( Qz l P ) (102)= x ◦ (χP ) ◦ ( Qz l P )= x ◦ (χP ) ◦ (QP χP ) ◦ (Qδ D P QP ) (128)= x ◦ (χP ) ◦ (χP QP ) ◦ (Qδ D P QP )(130)= x ◦ (χP ) ◦ ( Qε D P QP ) = x ◦ (χP ) ◦ (Qz r P )(102)= m A ◦ (xx) ◦ (Qz r P ) = m A ◦ (xA) ◦ (QP x) ◦ (Qz r P ) .Since Q preserves coequalizers we have(Q ̂Q, Ql) = Coequ Fun((QP x) ◦(Qz l P ) , (QP x) ◦ (Qz r P ) )so that there exists a functorial morphism σ A : Q ̂Q → A which satisfies (161). Nowwe want to show that σ A is A-bilinear that is the following equalities hold( )σ A ◦A µ Q ̂Q = m A ◦ ( Aσ A))σ A ◦(Qµ A Q b = m A ◦ ( σ A A ) .We computem A ◦ ( Aσ A) ◦ (AQl) (161)= m A ◦ (Am A ) ◦ (AxA)m A ass= m A ◦ (m A A) ◦ (AxA) (104)= m A ◦ (xA) ◦ (A µ Q P A )(161)= σ A ◦ (Ql) ◦ (A µ Q P A ) A µ Q= σ A ◦Since AQl is an epimorphism, we get that σ A ◦(A µ Q ̂Q( )A µ Q ̂Q)◦ (AQl) .m A ◦ ( σ A A ) ◦ (QlA) (161)= m A ◦ (m A A) ◦ (xAA)= m A ◦ (Am A ) ◦ (xAA) = x m A ◦ (xA) ◦ (QP m A ))(161)= σ A ◦ (Ql) ◦ (QP m A ) (150)= σ A ◦(Qµ A Q b ◦ (QlA)= m A ◦ ( Aσ A) . We compute