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

67and alsoϕ= ( RC C Ûɛ ) ◦ (RϕLD ϕ ) ◦ (RLηD ϕ ) ◦ (ηD ϕ )(R C Uγ C) ◦ d ϕ = ( R C Uγ C) ◦ ( R C Ûɛ ) ◦ (ηD ϕ )bɛmorph C A=(RC C Ûɛ ) ◦ ( R C Uγ C K ϕ D ϕ)◦ (ηDϕ )so thatdefK ϕ=(RC C Ûɛ ) ◦ (RϕLD ϕ ) ◦ (RLηD ϕ ) ◦ (ηD ϕ )(α C U ) ◦ d ϕ = ( R C Uγ C) ◦ d ϕ .Now, we will prove that the following diagram is commutative( HomC A Kϕ Y, ( ))X, C ba (X, C ρX ),Yρ XC UHom A (LY, X)Hom B(Y, Dϕ(X, C ρ X))a X,Y HomB (Y, RX) .Hom B(Y,d ϕ(X, C ρ X))In fact, for every ζ ∈ HomC A(Kϕ Y, ( X, C ρ X)), we have[HomB(Y, dϕ(X, C ρ X))◦ â(X, C ρ X ),Y](ζ)defba= Hom B(Y, dϕ(X, C ρ X))[(Dϕ ζ) ◦ (̂ηY )]= ( d ϕ(X, C ρ X))◦ (Dϕ ζ) ◦ (̂ηY )defd ϕ=(R C Ûɛ ( X, C ρ X))◦(ηDϕ(X, C ρ X))◦ (Dϕ ζ) ◦ (̂ηY )η= ( R C Ûɛ ( X, C ρ X))◦ (RLDϕ ζ) ◦ (RL̂ηY ) ◦ (ηY )defK ϕ=(R C Ûɛ ( X, C ρ X))◦(R C UK ϕ D ϕ ζ ) ◦ ( R C UK ϕ̂ηY ) ◦ (ηY )bɛ= ( R C Uζ ) ◦ ( R C ÛɛK ϕ Y ) ◦ ( R C UK ) ϕ̂ηY ◦ (ηY ) (Kϕ,Dϕ)= ( R C Uζ ) ◦ (ηY )and on the other hand(aX,Y ◦ C U ) (ζ) = a X,Y( CUζ ) defa= ( R C Uζ ) ◦ (ηY )so that, for every ( X, C ρ X)∈ C A we have( ( ))Hom B −, dϕ X, C ρ X ◦ â(X, C ρ X ),− = a X,− ◦ C U.( ( ))Since a X,− and â (X, C ρ X ),− are isomorphisms, we deduce that Hom B −, dϕ X, C ρ Xis ( mono. ) Applying the commutativity of this diagram in the particular case ofX, C ρ X = Kϕ Y, we get that(d ϕ K ϕ Y ) ◦ (̂ηY ) = Hom B (Y, d ϕ K ϕ Y ) (̂ηY )= Hom B (Y, d ϕ K ϕ Y ) ( ( ))â KϕY,Y IdKϕY= [ ] ( )Hom B (Y, d ϕ K ϕ Y ) ◦ â KϕY,Y IdKϕY= [ aC UK ϕY,Y ◦ C U ] ( )Id KϕY = (aLY,Y ) (C )UId KϕY= (a LY,Y ) ( IdC UK ϕY)= aLY,Y (Id LY ) = ηY