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

132= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (Qjj) ◦ ( Q∆ D)(67)= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP τ) ◦ (Qj)We computeσ A = µ B Q ◦ (A µ Q B ) ◦ ( σ A QB ) ◦ ( QP Qσ B) ◦ (QP τ) ◦ (Qj)(70)= µ B Q ◦ (A µ Q B ) ◦ ( σ A QB ) ◦ (QP Qu B ) ◦ (Qj)σ A = µ B Q ◦ (A µ Q B ) ◦ (AQu B ) ◦ ( σ A Q ) ◦ (Qj)Qis a bim= A µ Q ◦ ( Aµ B Q)◦ (AQuB ) ◦ ( σ A Q ) ◦ (Qj)Qis a mod= A µ Q ◦ ( σ A Q ) ◦ (Qj) (82)= µ B Q ◦ ( Qσ B) ◦ (Qj)(67)= µ B Q ◦ (Qu B ) ◦ ( Qε D) Qis a mod= Qε D .χ ◦ (δ C Q) =µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ (δ C Q)(147)= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (iCQ) ◦ ( ∆ C Q )= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (iiQ) ◦ ( ∆ C Q )(63)= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (τP Q) ◦ (iQ)(69)= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ (u A QP Q) ◦ (iQ)u A= µ B Q ◦ (A µ Q B ) ◦ (u A QB) ◦ ( Qσ B) ◦ (iQ)Qis a bim= A µ Q ◦ ( Aµ B Q)◦ (uA QB) ◦ ( Qσ B) ◦ (iQ)u A= A µ Q ◦ (u A Q) ◦ µ B Q ◦ ( Qσ B) ◦ (iQ)Qis a mod= µ B Q ◦ ( Qσ B) ◦ (iQ) (82)= A µ Q ◦ ( σ A Q ) ◦ (iQ)(63)= A µ Q ◦ (u A Q) ◦ ( ε C Q ) Qis a mod= ε C Q.□7.3. Constructing the functor ̂Q. Our next task is to construct a B-A-bimodulefunctor ̂Q.Proposition 7.6. Within the assumptions and notations of Theorem 6.29, definea functor ̂Q via the coequalizer DP QP (P x)◦(zl P )P A l ˆQ(P x)◦(z r P )Then there exists a unique functorial morphism ν ′ 0 : BP → ̂Q such that(149) ν ′ 0 ◦ (yP ) = l ◦ (P x) .