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

78Corollary 4.63 (Beck’s Precise Cotripleability Theorem). Let L : B → A be afunctor. Then L is comonadic if and only if1) L has a right adjoint R : A → B,2) for every ( X, LR ρ X)∈ LR A, there exist Equ B(ηRX, R LR ρ X)and L preservesthe equalizerEqu Fun((ηR) ◦( LRU ) , R LR Uγ LR)3) L reflects isomorphisms.In this case in B there exist equalizers of reflexive L-contractible equalizer pairsand L preserves them.Proof. Apply Theorem 4.62 to the case ϕ = Id LR .Lemma 4.64. Let (L, R) be an adjunction, where L : B → A and R : A → B, withunit η and counit ɛ. Then for every Y ∈ B,(LY, LRLY, LRLRLY, LηY, LηRLY, LRLηY, ɛLY, ɛLRLY ) is a contractible equalizerand in particular, for every Y ∈ BProof. Consider the following diagram(LY, LηY ) = Equ A (LηRLY, LRLηY ) .□and let us computeLY LηY LRLYɛLYLηRLYɛLRLYLRLηY LRLRLY(ɛLRLY ) ◦ (LηRLY ) = Id LRLY(ɛLY ) ◦ (LηY ) = Id LY(ɛLRLY ) ◦ (LRLηY ) = (LηY ) ◦ (ɛLY ) = Id LRLY(LηRLY ) ◦ (LηY ) = (LRLηY ) ◦ (LηY ) .Thus (LY, LRLY, LRLRLY, LηY, LηRLY, LRLηY, ɛLY, ɛLRLY ) is a contractibleequalizer for every Y ∈ B and by Proposition <strong>2.</strong>19 we get that (LY, LηY ) =Equ A (LηRLY, LRLηY ) .□Lemma 4.65. Let (L, R) be an adjunction where L : B → A and R : A → B, letC = ( C, ∆ C , ε C) be a comonad on a category A and let ϕ : LR = (LR, LηR, ɛ) →C = ( C, ∆ C , ε C) be a comonad morphism. Let K ϕ = Υ (ϕ) = (L, (ϕL) ◦ (Lη)) andC UK ϕ (f) = L (f) for every morphism f in B. For every Y ∈ B we have(55) (K ϕ Y, K ϕ ηY ) = EquC A (K ϕ ηRLY, K ϕ RLηY ) .Proof. By Lemma 4.64 we have that (LY, LηY ) = Equ A (LηRLY, LRLηY ). Leth : Z → K ϕ RLY = (LRLY, (ϕLRLY ) ◦ (LηRLY )) be a morphism in C A such thatThen(K ϕ RLηY ) ◦ h = (K ϕ ηRLY ) ◦ h.(56) (LRLηY ) ◦ (C Uh ) = (LηRLY ) ◦ (C Uh )