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

Proof. Let ( Q : B → A, C ρ Q)be a left C-comodule functor. Then, by Lemma 4.28,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 .Note that, since CQ preserves equalizers, by Lemma 4.18, Q = C U ◦ C Q preservesequalizers. Then, by Lemma 4.19, also C Q preserves equalizers. Conversely, ifH : B → C A is a functor preserving equalizers, we get that C U ◦ H : B → A.Moreover, by Lemma 4.20, C U preserves equalizers and thus also C U ◦ H preservesequalizers. Now, let us prove that C ν and C κ determine a bijective correspondencebetween C F and (C A ← B ) . We compute( Cκ ◦ C ν ) (( Q, C ρ Q))= C κ (C Q ) = (C U C Q, C Uγ C C Q ) = ( Q, C ρ Q).On the other hand we have( Cν ◦ C κ ) (H) = C ν ((C U ◦ H, C Uγ C H )) = C (C U ◦ H ) (41)= H.Theorem 4.37. Let C = ( C, ∆ C , ε C) be a comonad on a category A with equalizerssuch that C preserves equalizers. Let D = ( D, ∆ D , ε D) be a comonad on a categoryB with equalizers such that D preserves equalizers. Then there exists a bijectivecorrespondence between the following collections of data:( C F D C-D-bimodule functors Q : B → A such that CQ and QD preserve equalizersCA ← D B ) functors G : D B → C A preserving equalizersgiven byC ν D :C F D → (C A ← D B ) where C ν D (( Q, C ρ Q , ρ D Q))= C Q D59□C κ D :( CA ← D B ) → C F D where C κ D (G) = (C U ◦ G ◦ D F, C Uγ C G D F, C UGγ DD F ) .Proof. Let us consider a C-D-bicomodule functor ( Q : B → A, C ρ Q , ρQ) D such thatCQ and QD preserve equalizers. In particular, ( Q, ρQ) D is a right D-comodule functor,so that we can apply the map ν D : F D → ( A ← D B ) of Theorem 4.35 and weget a functor ν (( D Q, ρQ)) D = Q D : D B → A which preserves equalizers. By Proposition4.29, ( Q D , C ρ Q D)is a left C-comodule functor so that we can also apply themap C ν : C F → (C A ← B ) of Theorem 4.36 where the category B is D B. The mapC ν is defined by C ν (( Q D , C ρ Q D)) ( =CQ D) = C Q D : D B → C A and C Q D preservesequalizers. Conversely, let us consider a functor G : D B → C A which preservesequalizers. By Theorem 4.36, we get a left C-comodule functor given byC κ (G) = (C U ◦ G, C Uγ C G )where C U ◦ G : D B → A and C C UG preserves equalizers. By Lemma 4.18, alsoC U ◦ G : D B → A preserves equalizers. Thus, we can apply Theorem 4.35 and weget a right D-comodule functorκ D (C UG ) = (C UG D F , C UGγ DD F )where C UG D F : B → A is such that C UG D F D preserves equalizers. Clearly, sinceC UG preserves equalizers, D F is a right adjoint and C preserves equalizers by assumption,we deduce that also C C UG D F preserves equalizers. Now, we want to