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

2041) − ⊗ B Σ A preserves the equalizerHomC (Mod-A) ((Σ, ρ Σ ) , (X, x))i Hom A (Σ, X)x◦−(−⊗ A C)◦ρ Σ Hom A (Σ, X ⊗ A C) .2) can : Hom A ( B Σ A , −) ⊗ B Σ A → − ⊗ A C is a comonad isomorphism.Proof. Apply Theorem 4.53 to the adjunction (− ⊗ B Σ A , Hom A ( B Σ A , −)) .Theorem 9.7 ([GT, Theorem 3.2]). K ϕ : Mod-B → C (Mod-A) = Comod-C is anequivalence of categories if and only if1) − ⊗ B Σ A preserves the equalizerHomC (Mod-A) ((Σ, ρ Σ ) , (X, x))i Hom A (Σ, X)x◦−(−⊗ A C)◦ρ Σ Hom A (Σ, X ⊗ A C) .2) − ⊗ B Σ A reflects isomorphisms and3) can : Hom A ( B Σ A , −) ⊗ B Σ A → − ⊗ A C is a comonad isomorphism.Proof. Apply Theorem 4.55 to the adjunction (− ⊗ B Σ A , Hom A ( B Σ A , −)) .Let us now consider a particular case of the previous situation.Let C be an A-coring and let Σ be a right C-comodule. Set T = EndC (Mod-A) ((Σ, ρ Σ )).Then it is easy to check that (Σ, ρ Σ ) is a T -C-comodule. Following [Wis], we saythat Σ is a Galois C-comodule whenever can : Hom A ( T Σ A , −) ⊗ T Σ → − ⊗ A C isan isomorphism. The adjunction (C U, C F ) for C = (− ⊗ A C, − ⊗ A ∆, r ◦ (− ⊗ A ε))gives us the followingProposition 9.8. Let C be an A-coring and let Σ be a right C-comodule.T = EndC (Mod-A) ((Σ, ρ Σ )). Then the mapψL : Hom A ( T Σ A , L) → HomC (Mod-A)((Σ, ρΣ ) , C F L ) defined by settingψL (f) = (f ⊗ A C) ◦ ρ Σis an isomorphism whose inverse is defined by setting (ψL) −1 (h) = r L ◦ (L ⊗ A ε) ◦ h, for every L ∈ Mod-A. In this way we get a functorial isomorphismψ : Hom A ( T Σ A , −) → HomC (Mod-A)((Σ, ρΣ ) , C F ) .9.9. Note that, in particular, we have(ψA : Hom A ( T Σ A , A) → HomC (Mod-A) (Σ, ρΣ ) , C F A )whereso thatHomC (Mod-A)((Σ, ρΣ ) , C F A ) = HomC (Mod-A) ((Σ, ρ Σ ) , A ⊗ A C)and is defined by setting≃ HomC (Mod-A) ((Σ, ρ Σ ) , C)ψA : Hom A ( T Σ A , A) → HomC (Mod-A) ((Σ, ρ Σ ) , C)[ψA (f)] (t) = [l C ◦ (f ⊗ A C) ◦ ρ Σ ] (t) = f (t 0 ) t 1 .□□Set