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

= (CCɛ) ◦ (CβR) ◦ (CɛLR) ◦ (βRLR) ◦ (LηR)β= (CCɛ) ◦ (CβR) ◦ (CɛLR) ◦ (CLηR) ◦ (βR) = (CCɛ) ◦ (CβR) ◦ (βR)(L,β)= (CCɛ) ◦ ( ∆ C LR ) ◦ (βR) =∆ C= ∆ C ◦ (Cɛ) ◦ (βR) = ∆ C ◦ Λ (β)andε C ◦ Λ (β) = ε C ◦ (Cɛ) ◦ (βR) εC = ɛ ◦ ( ε C LR ) ◦ (βR) (L,β)= ɛ.Therefore we deduce that Λ (β) ∈ M. Let now ϕ ∈ M and let us calculateΞΘ (ϕ) = (ɛC) ◦ (LRϕ) ◦ (LηR) ɛ = ϕ ◦ (ɛLR) ◦ (LηR) = ϕ.Let now α ∈ R and let us calculateΘΞ (α) = (RΞ (α)) ◦ (ηR) = (RɛC) ◦ (RLα) ◦ (ηR) (RɛC) ◦ (ηRC) ◦ α = α.Let now ϕ ∈ M and let us calculateΛΓ (ϕ) = (Cɛ) ◦ (Γ (ϕ) R) = (Cɛ) ◦ (ϕLR) ◦ (LηR) ϕ = ϕ ◦ (LRɛ) ◦ (LηR) = ϕ.Let now β ∈ L and let us calculateΓΛ (β) = (Λ (β) L) ◦ (Lη) = (CɛL) ◦ (βRL) ◦ (Lη) β = (CɛL) ◦ (CLη) ◦ β = β.Theorem 4.43 ([D, Theorem II.<strong>1.</strong>1] and [GT, Theorem <strong>1.</strong>2]). Let (L, R) be an adjunctionwhere L : B → A and R : A → B and let C = ( C, ∆ C , ε C) be a comonadon a category A. There exists a bijective correspondence between the following collectionsof data:K Functors K : B → C A such that C U ◦ K = L,M comonad morphisms ϕ : LR = (LR, LηR, ɛ) → C = ( C, ∆ C , ε C)given byΨ : K → M where Ψ (K) = (Cɛ) ◦ ([C U ( γ C K )] R )Υ : M → K where Υ (ϕ) (Y ) = (LY, (ϕLY ) ◦ (LηY )) and Υ (ϕ) (f) = L (f) .Proof. By Corollary 4.25, there exists a bijective correspondence between K and thecollection L of functorial morphisms β : L → CL such that (L, β) is a left comodulefunctor for the comonad C given byΦ : K → L where Φ (K) = C U ( γ C K ) : L → CLΩ : L → K where Ω (β) (Y ) = (LY, βY ) and Ω (β) (f) = L (f) .By Proposition 4.42, there exists a bijective correspondence between L and thecollection M of comonad morphisms ϕ : LR = (LR, LηR, ɛ) → C = ( C, ∆ C , ε C)given byWe computeΛ : L → M where Λ (β) = (Cɛ) ◦ (βR)Γ : M → L where Γ (ϕ) = (ϕL) ◦ (Lη) .(Λ ◦ Φ) (K) = (Cɛ) ◦ ([C U ( γ C K )] R ) = Ψ (K)63□