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

Theorem 8.6. Within the assumptions and notations of Theorem 6.29, we have afunctorial isomorphism B ̂QAA Q B∼ = B B.Proof. In Proposition 8.5 we have proved the existence of a functorial morphism α :B B U → ̂Q(AA Q B such that ̂QAA Q B , α)= Coequ Fun (m BB U, B B Uλ B. ) . By Proposition3.13 and Proposition 3.14 also ( B U, ( B Uλ B )) = Coequ Fun (m BB U, B B Uλ B. ). Inview of uniqueness of a coequalizer up to isomorphisms, there exists a functorialisomorphismNow sinceβ : ̂Q AA Q B = B U B ̂QAA Q B → B U such that β ◦ α = B Uλ B .( B Uλ B ) ◦ (m BB U) = ( B Uλ B ) ◦ (B B Uλ B )and since β ◦ α = B Uλ B , by applying (177) where ”ĥ” = β and ”h” = BUλ B , wededuce that)β ◦(pb Q AQ B ◦ (l A U A Q B ) ◦ (P Ap Q ) = ( B Uλ B ) ◦ (y B U) ◦ ( P A µ Q BU )equivalentlyi.e.(189) β ◦)β ◦(pb Q AQ B ◦ (l A U A Q B ) ◦ (P Ap Q ) ◦ (P xQ B U)= ( B Uλ B ) ◦ (y B U) ◦ ( P A µ Q BU ) ◦ (P xQ B U)(101)= ( B Uλ B ) ◦ (y B U) ◦ (P χ B U)(pb Q AQ B)◦ (l A U A Q B ) ◦ (P Ap Q ) ◦ (P xQ B U) = ( B Uλ B ) ◦ (y B U) ◦ (P χ B U) .Recall ( that, in view of Proposition 3.13, ( B U, B Uλ B ) is an B-left module functor.Also ̂QAA Q B , B µb QA A B)Q is an B-left module functor (see proof of Proposition3.30 and Lemma 3.17) where B µb QA= B Uλ BB ̂QA : B ̂Q A → ̂Q A .Now we want toprove that β lifts to a functorial morphism B ̂QAA Q B → B B i.e. thatβ :(̂QAA Q B , B µb QA AQ B)→ ( B U, B Uλ B )is a morphism of B-left module functors. Thus we have to proveWe calculate( B Uλ B ) ◦ (Bβ) = β ◦ ( B µb QA AQ B )(y ̂Q)◦ (P Ql) ◦ (P QP x)B µb Q◦ (Bl) ◦ (yP A) ◦ (P QP x) = B µb Q◦((149)= B µb Q◦ y ̂Q)◦ (P Qν 0) ′ ◦ (P QyP ) = y B µb Q◦ (Bν 0) ′ ◦ (yBP ) ◦ (P QyP )(151)= ν ′ 0 ◦ (m B P ) ◦ (yBP ) ◦ (P QyP ) = ν ′ 0 ◦ (m B P ) ◦ (yyP )(109)= ν ′ 0 ◦ (yP ) ◦ (P χP ) (149)= l ◦ (P x) ◦ (P χP )161