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

58F D right D-comodule functors Q : B → A such that QD preserves equalizers.(A ← D B ) functors G : D B → A preserving equalizersgiven byν D : F D → ( A ← D B ) where ν (( )) D Q, ρ D Q = QD(κ D : A ← D B ) → F D where κ D (G) = ( G D F , Gγ DD F )where Q D is uniquely determined by ( Q D , ι Q) = Equ Fun(ρDQ D U, Q D Uγ D) .Proof. Let Q : B → A be a right D-comodule functor. Then we can considerQ D : D B → A defined by (32) as(Q D , ι Q) (= Equ Fun ρD DQ U, Q D Uγ D) .Since by assumption QD preserves equalizers, by Lemma 4.18 also Q preservesequalizers. Moreover, since D preserves equalizers, by Lemma 4.20 also the functorD U preserves equalizers. Thus both QD D U and Q D U preserve equalizers. ByCorollary <strong>2.</strong>14 we get that also Q D : D B → A preserves equalizers.Conversely, let us consider a functor G : D B → A that preserves equalizers. ByProposition 4.33 we can consider the right D-comodule functor defined as followsQ = G ◦ D F and let ρ D Q = Gγ DD F .Since D F is right adjoint to D U in particular D F preserves equalizers and since byassumption G preserves equalizers, we get that also Q = G ◦ D F preserves equalizersand so does QD.Now, we want to prove that ν D and κ D determine a bijective correspondencebetween F D and ( A ← D B ) . Let us start with a right D-comodule functor(Q : B → A, ρDQ). Then we have(κ D ◦ ν D) (( Q, ρ D Q))= κD ( Q D) = ( Q DD F , Q D γ DD F )= ( Q DD F , ρ D Q DD F) (39)= ( Q, ρ D Q).Moreover we have(ν D ◦ κ D) (G) = ν D (( G D F , Gγ DD F )) = ( G D F ) D (40)= G.Theorem 4.36. Let C = ( C, ∆ C , ε C) be a comonad on a category A with equalizerssuch that C preserves equalizers. Then there exists a bijective correspondencebetween the following collections of data:C F left C-comodule functors Q : B → A such that CQ preserves equalizers( CA ← B ) functors H : B → C A preserving equalizers□given byC ν :C κ :C F → (C A ← B ) where C ν (( ))Q, C ρ Q = C Q( CA ← B ) → C F where C κ (H) = (C U ◦ H, C Uγ C H )where C Q : B → C A is the functor defined in Lemma 4.28.