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

70In particular(49) d ϕ(CX, ∆ C X ) = αX.Corollary 4.50. In the setting of Proposition 4.47, assume that, for every ( X, C ρ X)∈C A, there exists Equ B(αX, R C ρ X). Then for every X ∈ A we haveC Ûɛ ( CX, ∆ C X ) = ϕXand henceC Ûɛ C F = ϕwhere ̂ɛ is the counit of the adjunction (K ϕ , D ϕ ).Proof. Let us calculateC Ûɛ ( CX, ∆ C X ) (47)= (ɛCX) ◦ ( Ld ϕ(CX, ∆ C X ))(48)= (ɛCX) ◦ (LαX) = Ξ (α) (X) = ϕX.Corollary 4.5<strong>1.</strong> In the setting of Proposition 4.47, assume that, for every ( X, C ρ X)∈C A, there exists Equ B(αX, R C ρ X). Then, the functor Dϕ is full and faithful if andonly if ̂ɛ is a functorial isomorphism.Proof. By Proposition 4.47, (K ϕ , D ϕ ) is an adjunction with counit ̂ɛ : K ϕ D ϕ → C A.Then we can apply Proposition <strong>2.</strong>3<strong>2.</strong>□Lemma 4.52 ([GT, Lemma <strong>2.</strong>5]). In the setting of Proposition 4.47, assume that, forevery ( X, C ρ X)∈ C A, there exists Equ B(αX, R C ρ X). Then, for every(X, C ρ X)∈C A the following diagram□LD ϕ(X, C ρ X) C Ubɛ(X, C ρ X)C U ( X, C ρ X)Ld ϕ(X, C ρ X) LR C U ( )X, C ϕ C U(X, C ρ X)ρ XC U C γ(X, C ρ X)C C U ( )X, C ρ XLα C U(X, C ρ X) LR C U C γ(X, C ρ X) ∆ C C U(X, C ρ X) C C U C γ(X, C ρ X)LRC C U ( )X, C ϕC C U(X, C ρ X)ρ X CC C U ( )X, C ρ Xserially commutes. Therefore we get( CUγ C) ◦ (C Ûɛ ) = ( ϕ C U ) ◦ (Ld ϕ ) andProof. Let us compute(∆C C U ) ◦ ( ϕ C U ) = ( ϕC C U ) ◦ ( Lα C U ) .C ρ X ◦ C Ûɛ ( X, C ρ X) (47)= C ρ X ◦ (ɛX) ◦ ( Ld ϕ(X, C ρ X))ϕmorphcomonads=C ρ X ◦ ( ε C X ) ◦ (ϕX) ◦ ( Ld ϕ(X, C ρ X))ε C = ( ε C CX ) ◦ ( C C ρ X)◦ (ϕX) ◦(Ldϕ(X, C ρ X))ϕ= ( ε C CX ) ◦ (ϕCX) ◦ ( ) ( ( ))LR C ρ X ◦ Ldϕ X, C ρ X