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

134Assume now that there is another morphism t : ̂Q → X such that ξ = t ◦ ν ′ 0. Thenwe havet ◦ l ◦ (P x) (149)= t ◦ ν ′ 0 ◦ (yP ) = ξ ◦ (yP ) = ν ◦ ν ′ 0 ◦ (yP ) (149)= ν ◦ l ◦ (P x) .Since l ◦ (P x) is an epimorphism, we deduce that t = ν.(2) We want to equip ̂Q with the structure of a B-A-bimodule functor. To beginwith, let us prove a number of equalities. Let us calculateso thatχ ◦ ( Qz l) = χ ◦ (QP χ) ◦ (Qδ D P Q) (98)= χ ◦ (χP Q) ◦ (Qδ D P Q)(105)= χ ◦ ( Qε D P Q ) = χ ◦ (Qz r )(153) χ ◦ ( Qz l) = χ ◦ (Qz r ) .Let(154) b = m A ◦ (xA) .Thenso thatx ◦ (χP ) (102)= m A ◦ (xx) = m A ◦ (xA) ◦ (QP x)(155) x ◦ (χP ) = b ◦ (QP x) .We haveand hence(P χ) ◦ ( z l P Q ) = (P χ) ◦ (P χP Q) ◦ (δ D P QP Q)(98)= (P χ) ◦ (P QP χ) ◦ (δ D P QP Q)δ D= (P χ) ◦ (δ D P Q) ◦ (DP χ) = z l ◦ (DP χ)y ◦ (P χ) ◦ ( z l P Q ) = y ◦ z l ◦ (DP χ) ycoequ= y ◦ z r ◦ (DP χ) = y ◦ ( ε D P Q ) ◦ (DP χ)so that we getε D = y ◦ (P χ) ◦ ( ε D P QP Q ) = y ◦ (P χ) ◦ (z r P Q)(156) y ◦ (P χ) ◦ ( z l P Q ) = y ◦ (P χ) ◦ (z r P Q) .From previous equalities, it follows thatl ◦ (P b) ◦ ( z l P A ) ◦ (DP QP x) = zll ◦ (P b) ◦ (P QP x) ◦ ( z l P QP )(155)= l ◦ (P x) ◦ (P χP ) ◦ ( z l P QP ) (149)= ν ′ 0 ◦ (yP ) ◦ (P χP ) ◦ ( z l P QP )(156)= ν ′ 0 ◦ (yP ) ◦ (P χP ) ◦ (z r P QP ) (149)= l ◦ (P x) ◦ (P χP ) ◦ (z r P QP )(155)= l ◦ (P b) ◦ (P QP x) ◦ (z r P QP ) zr= l ◦ (P b) ◦ (z r P A) ◦ (DP QP x) .Since DP QP x is an epimorphism, we obtain(157) l ◦ (P b) ◦ ( z l P A ) = l ◦ (P b) ◦ (z r P A)