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

52where ε C : C = C U C F → A is also the counit of the adjunction (C U, C F ) andγ D : D B → D F D U is the unit of the adjunction (D U, D F ) . Let now definethat isΞ def= D Uξ : D U ˜T C F = T C U C F = T C → D U D F T = DTΞ = D Uξ = (D U D F T ε C) ◦( )D Uγ D ˜T C F .Dually to Proposition 3.24 you can prove that Ξ is a functorial morphism satisfying(∆ D T ) ◦ Ξ = (DΞ) ◦ (ΞC) ◦ ( T ∆ C) and ( ε D T ) ◦ Ξ = T ε C .Conversely, let Ξ be a functorial morphism satisfying ( ∆ D T ) ◦ Ξ = (DΞ) ◦ (ΞC) ◦(T ∆C ) and ( ε D T ) ◦ Ξ = T ε C . We define ˜T : C A → D B by setting, for every(X, C ρ X)∈ C A,˜T (( X, C ρ X))=(T X, (ΞX) ◦(T C ρ X))and for every f : ( X, C ρ X)→(Y, C ρ Y)∈ C A,˜T (f) = T (f) .Dually to Proposition 3.24 you can prove that ˜T is a functor between C A → D B whichlifts T and that a : F → M and b : M → F define a bijective correspondence. □Corollary 4.24. Let X , A be categories and let C = ( C, ∆ C , ε C) be a comonad ona category A and let F : X → A be a functor. Then there is a bijection between thefollowing collections of data:F Functors C F : X → C A such that C U C F = F ,G Left C-comodule coactions C ρ F : F → CFgiven byα : F → G where α (C F ) = C Uγ C C F : F → CFβ : G → F where C Uβ (C ρ F)= F and C Uγ C β (C ρ F)= C ρ F i.e.β : G → F where β (C ρ F)(X) =(F X, C ρ F X ) and β (C ρ F)(f) = F (f) .Proof. Apply Proposition 4.23 to the case A = X , B = A, C = Id X , D = C. Then˜T = C F is the lifting of F and Ξ = C ρ F : F → CF satisfies ( ∆ C F ) ◦ C ρ F =(C C ρ F)◦ C ρ F and ( ε C F ) ◦ C ρ F = F that is ( F, C ρ F)is a left C-comodule functor. □Corollary 4.25. Let (L, R) be an adjunction where L : B → A, R : A → B andlet C = ( C, ∆ C , ε C) be a comonad on a category A. Then there exists a bijectivecorrespondence between the following collections of data:K Functors K : B → C A such that C U ◦ K = L,L Functorial morphism β : L → CL such that (L, β) is a left comodule functorfor the comonad Cgiven byΦ : K → L where Φ (K) = C U ( γ C K ) : L → CLΩ : L → K where Ω (β) (Y ) = (LY, βY ) and C UΩ (β) (f) = L (f) .