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

Proof. Apply Corollary 4.24 to the case ”F ” = L : B → A where (L, R) is anadjunction and C = ( C, ∆ C , ε C) a comonad on A.□Proposition 4.26. Let C = ( C, ∆ C , ε C) be a comonad on a category A and letD = ( D, ∆ D , ε D) be a comonad on a category B. Let T : A → B be a functor,let ˜T : C A → D B be a lifting of T (i.e.D U ˜T = T C U) and let Ξ : T C → DTas in Proposition 4.23. Then Ξ is an isomorphism if and only if ξ = (D ( )F T ε C) ◦γ D ˜T C F : ˜T C F → D F T is an isomorphism.Proof. By construction in Proposition 4.23 we have that Ξ = D Uξ. Assume thatΞ is an isomorphism. Since, by Proposition 4.17, D U reflects isomorphisms, ξ :˜T C F → D F T is an isomorphism. Conversely, assume that ξ : ˜T C F → D F T is anisomorphism. Then D Uξ is also an isomorphism.□Corollary 4.27. Let (L, R) be an adjunction where L : B → A and R : A → Band let C = ( C, ∆ C , ε C) be a comonad on B. Let K : B → C A be a functor suchthat C U ◦ K = L and let (L, β) be a left C-comodule functor as in Corollary 4.25.Then β is an isomorphism if and only if γ C K : K → C F L is an isomorphism.Proof. Apply Proposition 4.26 with T = L so that the categories A and B areinterchanged, C = Id B and D = C. Then ˜T = K is the lifting of L and Ξ = β : L →CL, given by β = C Uξ = C Uγ C K.□Lemma 4.28. Let C = ( C, ∆ C , ε C) be a comonad over a category A with equalizers.Let Q : B → A be a left C-comodule functor with functorial morphisms C ρ Q : Q →CQ. Then there exists a unique functor C Q : B → C A such thatC U C Q = Q and C Uγ C C Q = C ρ Q .Moreover if ψ : Q → T is a functorial morphism between left C-module functors andψ satisfiesC ρ Q ◦ (Cψ) = ψ ◦ (C ρ T)then there is a unique functorial morphism C ψ : C Q → C T such thatC U C ψ = ψ.Proof. Corollary 4.24 applied to the case where F = Q and C ρ F = C ρ Q gives us thefirst statement. Let B ∈ B. Then we have( Cρ Q B ) ◦ (CψB) = (ψB) ◦ (C ρ T B )which means that ψB yields a morphism C ψB in C A.Proposition 4.29. Let C = ( C, ∆ C , ε C) be a comonad over a category A and letD = ( D, ∆ D , ε D) be a comonad over a category B. Assume that both A and B haveequalizers and that C preserves equalizers. Let Q : B → A be a functor and letC ρ Q : Q → CQ and ρ D Q : Q → QD be functorial morphisms. Assume that C ρ Q iscoassociative and counital and that ( CρQ) D ◦ C ρ Q = (C ρ Q D ) ◦ ρ D Q . Set((32)Q D , ι Q) (= Equ Fun ρD DQ U, Q D Uγ D) .53□