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

74we have that Ld ϕ K ϕ Y is mono and hence we obtainso that L̂η is a functorial isomorphism.(L̂ηY ) ◦ (ɛLY ) ◦ (Ld ϕ K ϕ Y ) = LD ϕ K ϕ YDefinition 4.56. Let ( L, C ρ L)be a left comodule functor for a comonad C =(C, ∆ C , ε C) such that L has a right adjoint R. Then we can consider a canonicalcomonad morphismcan := (Cɛ) ◦ (C ρ L R ) : LR → Cwhere ɛ denotes the counit of the adjunction (L, R) . A left C-Galois functor is aleft C-comodule functor ( L, C ρ L)with a right adjoint R such that can is a comonadisomorphism.Corollary 4.57. Let ( L, C ρ L)be a left C-Galois comodule functor such that Lpreserves equalizers, L reflects isomorphisms and let C = ( C, ∆ C , ε C) be a comonadon A. Assume that, for every ( X, C ρ X)∈ C A, there exists Equ B(αX, R C ρ X)whereα = (Rcan) ◦ (ηR) where R is the right adjoint of L and η is the unit of theadjunction. Then we can consider the functor K can : B → C A. Then the functorK can is an equivalence of categories.Proof. We can apply Theorem 4.55 to the case ϕ = can.Theorem 4.58 (Beck’s Theorem). Let (L, R) be an adjunction where L : B → A andR : A → B. Let α = Θ (Id LR ) = ηR and assume that, for every ( X, LR ρ X)∈ LR A,there exists Equ B(ηRX, R LR ρ X). Then we can consider the functor K = Υ (IdLR ) :B → LR A and its right adjoint D : LR A → B. The functor K is an equivalence ofcategories if and only if1) L preserves the equalizer2) L reflects isomorphisms.(D, d) = Equ Fun(ηR LR U, R LR Uγ LR) .Proof. Apply Theorem 4.55 taking ϕ = Id LR and thus α = Θ (Id LR ) = ηR.Definition 4.59. Let C = ( C, ∆ C , ε C) be a comonad on the category A and letL : B → A. The functor L is called ϕ-comonadic if it has a right adjoint R : A → Bfor which there exists ϕ : LR → C a comonad morphism such that the functorK ϕ = Υ (ϕ) : B → C A is an equivalence of categories with D ϕ : C A → B which isright adjoint.Definition 4.60. Let L : B → A be a functor. The functor L is called comonadicif it has a right adjoint R : A → B such that the functor K = Υ (Id LR ) : B → LR Ais an equivalence of categories with right adjoint D : LR A → B.Lemma 4.6<strong>1.</strong> Let L : B → A be a ϕ-comonadic functor and let(53) X ′ d 0 Xd 1□□□