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

are uniquely determined by(102) x ◦ (χP ) = m A ◦ (xx)and(103) u A ◦ ε C = x ◦ δ C .Finally ( Q, A µ Q)is a left A-module functor.Proof. We haveHenceχ ◦ ( w l Q ) = χ ◦ (χP Q) ◦ (QP δ C Q) (98)= χ ◦ (QP χ) ◦ (QP δ C Q)By Lemma <strong>2.</strong>9, we have that99= χ ◦ ( QP ε C Q ) = χ ◦ (w r Q) .χ ◦ ( w l Q ) = χ ◦ (w r Q) .(AQ, xQ) = Coequ Fun(w l Q, w r Q )and hence there exists a unique functorial morphism A µ Q : AQ → Q which fulfils(101). We computex ◦ ( A µ Q P ) ◦ (Aw l ) ◦ (xQP C) x = x ◦ ( A µ Q P ) ◦ (xQP ) ◦ ( QP w l)(101)= x ◦ (χP ) ◦ ( QP w l) = x ◦ (χP ) ◦ (QP χP ) ◦ (QP QP δ C )(98)= x ◦ (χP ) ◦ (χP QP ) ◦ (QP QP δ C )χ= x ◦ (χP ) ◦ (QP δ C ) ◦ (χP C) = x ◦ w l ◦ (χP C)xcoequ= x ◦ w r ◦ (χP C) = x ◦ ( QP ε C) ◦ (χP C)χ= x ◦ (χP ) ◦ ( QP QP ε C) (101)= x ◦ ( A µ Q P ) ◦ (xQP ) ◦ ( QP QP ε C)x= x ◦ ( A µ Q P ) ◦ ( AQP ε C) ◦ (xQP C) = x ◦ ( A µ Q P ) ◦ (Aw r ) ◦ (xQP C)so that we getx ◦ ( A µ Q P ) ◦ (Aw l ) ◦ (xQP C) = x ◦ ( A µ Q P ) ◦ (Aw r ) ◦ (xQP C)and since xQP C is an epimorphism we deduce thatx ◦ ( A µ Q P ) ◦ (Aw l ) = x ◦ ( A µ Q P ) ◦ (Aw r ).By Corollary <strong>2.</strong>12, A preserves coequalizers so that we get(AA, Ax) = Coequ Fun(Aw l , Aw r) .Hence there exists a unique functorial morphism m A : AA → A such that(104) m A ◦ (Ax) = x ◦ ( A µ Q P )or equivalentlym A ◦ (xx) = m A ◦ (Ax) ◦ (xQP ) = x ◦ ( A µ Q P ) ◦ (xQP ) (101)= x ◦ (χP ) .103