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

114= A µ Q ◦ ( Aε C Q ) ◦ (AC A µ Q )◦ ( ACAε C Q ) ◦ ( ACcocan −11 Q ) ◦ ( cocan −11 QP Q )= A µ Q ◦ (A A µ Q )◦ ( Aε C AQ ) ◦ ( ACAε C Q ) ◦ ( ACcocan −11 Q ) ◦ ( cocan −11 QP Q )= A µ Q ◦ (m A Q)◦ ( Aε C AQ ) ◦ ( ACAε C Q ) ◦ ( ACcocan −11 Q ) ◦ ( cocan −11 QP Q )= A µ Q ◦ (m A Q)◦ ( AAε C Q ) ◦ ( Aε C CAQ ) ◦ ( ACcocan −11 Q ) ◦ ( cocan −11 QP Q )= A µ Q ◦ ( Aε C Q ) ◦ (m A CQ) ◦ ( Acocan −11 Q ) ◦ ( Aε C QP Q ) ◦ ( cocan −11 QP Q )(131)= A µ Q ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ ( A µ Q P Q)◦ ( Aε C QP Q ) ◦ ( cocan −11 QP Q )Note that we haveso we have= χ ◦ (χP Q) .(cocan 1 ) ◦ (u A C) = (A µ Q P ) ◦ (Aδ C ) ◦ (u A C)= (A µ Q P ) ◦ (u A QP ) ◦ δ C A µ Q unital= δ C(132) (cocan 1 ) ◦ (u A C) = δ C .Now we computeand so we getWe haveχ ◦ (δ C Q) = A µ Q ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (δ C Q)(132)= A µ Q ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (cocan 1 Q) ◦ (u A CQ)= A µ Q ◦ ( Aε C Q ) ◦ (u A CQ) = A µ Q ◦ (u A Q) ◦ ( ε C Q ) ( A µ Q unital)=χ ◦ (δ C Q) = ( ε C Q ) .(Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) ◦ cocan 2= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) ◦ (A µ Q D ) ◦ ( Aρ D Q= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (A µ Q P Q ) ◦ (AQδ D ) ◦ ( Aρ D Q(127)= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (A µ Q P Q ) ◦ (Aδ C Q) ◦ ( A C ρ Q)= ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (cocan 1 Q) ◦ ( A C ρ Q)= ( Aε C Q ) ◦ ( A C ρ Q) C ρ Q counital= AQSince cocan 2 is an isomorphism, we have that(133) cocan 2 ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D ) = QD.Finally we computeρ D Q counital=χ ◦ (Qδ D ) = A µ Q ◦ ( Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D )))(ε C Q )A µ Q ◦ ( AQε D) ◦ ( (AρQ) D ◦ Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D )= ( Qε D) ◦ ( A µ Q D) ◦ ( ) (Aρ D Q ◦ Aε C Q ) ◦ ( cocan −11 Q ) ◦ (Qδ D )