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

133Moreover ( )( ( ̂Q, ν′0 = Coequ ) Fun (yP ) ◦ P wl, (yP ) ◦ (P w r ) )The functor ̂Q can be equipped with the structure of a B-A-bimodule functor(̂Q, µ A b Q, B µb Q)where µ A b Qand B µb Qare uniquely defined by(150) µ A b Q◦ (lA) = l ◦ (P m A )and(151)Proof. By construction we haveBy Lemma <strong>2.</strong>9, we haveB µb Q◦ (Bν ′ 0) = ν ′ 0 ◦ (m B P ) .l ◦ (P x) ◦ ( z l P ) = l ◦ (P x) ◦ (z r P ) .(BP, yP ) = Coequ Fun(z l P, z r P ) .By the universality of coequalizers, there exists a unique functorial morphism ν ′ 0 :BP → ̂Q which fulfils (149). Let us prove that(̂Q, ν′0)= Coequ Fun((yP ) ◦(P wl ) , (yP ) ◦ (P w r ) ) . We haveν ′ 0 ◦ (yP ) ◦ ( P w l) (149)= l ◦ (P x) ◦ ( P w l) defx= l ◦ (P x) ◦ (P w r )(149)= ν ′ 0 ◦ (yP ) ◦ (P w r ) .Let now ξ : BP → X be a morphism such that ξ ◦ (yP ) ◦ ( P w l) = ξ ◦ (yP ) ◦ (P w r ).Since P preserves coequalizers, we have(P A, P x) = Coequ Fun(P w l , P w r) .By universality of coequalizers there exists a unique functorial morphism ν 0 : P A →X such that(152) ν 0 ◦ (P x) = ξ ◦ (yP ) .We computeν 0 ◦ (P x) ◦ ( z l P ) (152)= ξ ◦ (yP ) ◦ ( z l P )= ξ ◦ (yP ) ◦ (z r P ) (152)= ν 0 ◦ (P x) ◦ (z r P ) .( )By the universality of the coequalizer ̂Q, l , there exists a unique functorial morphismν : ̂Q → X such thatWe computeSince yP is epi, we getν ◦ l = ν 0 .ν ◦ ν ′ 0 ◦ (yP ) (149)= ν ◦ l ◦ (P x) = ν 0 ◦ (P x) (152)= ξ ◦ (yP ) .ξ = ν ◦ ν ′ 0.