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

155holds where ν ′ 0 is defined in (149) .Proof. We computeν ′ 0 ◦ (u B P ) ◦ ( ε D P ) ◦ ( DP ε C) (110)= ν ′ 0 ◦ (yP ) ◦ (δ D P ) ◦ ( DP ε C)δ D= ν ′ 0 ◦ (yP ) ◦ ( P QP ε C) ◦ (δ D P C)(149)= l ◦ (P x) ◦ ( P QP ε C) ◦ (δ D P C) = l ◦ (P x) ◦ (P w r ) ◦ (δ D P C)xcoequ= l ◦ (P x) ◦ ( P w l) ◦ (δ D P C) = l ◦ (P x) ◦ (P χP ) ◦ (P QP δ C ) ◦ (δ D P C)δ D= l ◦ (P x) ◦ (P χP ) ◦ (δ D P QP ) ◦ (DP δ C )(149)= ν ′ 0 ◦ (yP ) ◦ (P χP ) ◦ (δ D P QP ) ◦ (DP δ C ) = ν ′ 0 ◦ (yP ) ◦ (z l P ) ◦ (DP δ C )ycoequ= ν ′ 0 ◦ (yP ) ◦ (z r P ) ◦ (DP δ C ) = ν ′ 0 ◦ (yP ) ◦ (ε D P QP ) ◦ (DP δ C )(149)= l ◦ (P x) ◦ (ε D P QP ) ◦ (DP δ C ) = l ◦ (P x) ◦ (P δ C ) ◦ (ε D P C)(103)= l ◦ (P u A ) ◦ ( P ε C) ◦ (ε D P C) εD = l ◦ (P u A ) ◦ ( ε D P ) ◦ (DP ε C ).Since ( ε D P ) ◦ ( DP ε C) is an epimorphism (recall that both ε D and ε C are coequalizers),we conclude.□Proposition 8.5. Within the assumptions and notations of Theorem 6.29, thereexists 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 ) . Moreover for every morphism h : B B U →X such thath ◦ (m BB U) = h ◦ (B B Uλ B )if ĥ : ̂Q AA Q B → X is the unique morphism such that ĥ ◦ α = h, we have that)(177) ĥ ◦(pb Q AQ B ◦ (l A U A Q B ) ◦ (P Ap Q ) = h ◦ (y B U) ◦ ( P A µ Q BU ) .Proof. Let us prove that)(178)(pb Q AQ B ◦ (l A U A Q B ) ◦ (P Ap Q ) ◦ (P u A Q B U) ◦ (P χ B U))=(pb Q AQ B ◦ (l A U A Q B ) ◦ (P Ap Q ) ◦ (P xQ B U) .Using Proposition 8.1, we compute) (pb Q AQ B ◦ (l A U A Q B ) ◦ (P Ap Q ) ◦ (P u A Q B U) ◦ (P χ B U))u=A(pb Q AQ B ◦ (l A U A Q B ) ◦ (P Ap Q ) ◦ (P Aχ B U) ◦ (P u A QP Q B U)(101)=(174)=) (pb Q AQ B ◦ (l A U A Q B ) ◦ (P Ap Q ) ◦ ( P A A µ QB U ) ◦ (P AxQ B U)◦ (P u A QP Q B U)) (pb Q AQ B ◦ (l A U A Q B ) ◦ ( )P A A µ QB ◦ (P AApQ ) ◦ (P AxQ B U)◦ (P u A QP Q B U)