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

60prove that C ν D : C F D → (C A ← D B ) and C κ D : (C A ← D B ) → C F D determine abijection. We have(Cκ D ◦ C ν D) (( ))Q, C ρ Q , ρ D Q = C κ D (C Q D)= (C U ◦ C Q D ◦ D F , C Uγ C C Q DD F , C U C Q D γ DD F ) = ( Q, C Uγ C C Q, Q D γ DD F )= ( ) ( )Q, C ρ Q , ρ D Q DD F = Q, C ρ Q , ρ D Qand( Cν D ◦ C κ D) (G) = C ν D ((C U ◦ G ◦ D F , C Uγ C G D F , C UGγ DD F ))= C (C U ◦ G ◦ D F ) D=C ((C U ◦ G ◦ D F ) D )(40)= C (C U ◦ G ) (41)= G.Proposition 4.38. Let C = ( C, ∆ C , ε C) be a comonad over a category A with equalizersand assume that C preserves equalizers. Let D = ( D, ∆ D , ε D) be a comonadover a category B with equalizers and let Q : B → A be a C-D-bicomodule functor.Then there exists a unique lifted functor C Q D : D B → C A such thatC U C Q DD F = Q.Proof. By Proposition 4.30 there exists a unique functor C Q D : D B → C A such thatC U C Q D = Q D . Now, by Proposition 4.32 we also get that Q DD F = Q so that weobtainC U C Q DD F = Q.Corollary 4.39. Let C = ( C, ∆ C , ε C) be a comonad over a category A with equalizersand assume that C preserves equalizers and let Q : A → A be a C-bicomodulefunctor. Then there exists a unique lifted functor C Q C : C A → C A such thatC U C Q C C F = Q.Proof. We can apply Proposition 4.38 to the case D = C and B = A.Proposition 4.40. Let C = ( C, ∆ C , ε C) be a comonad over a category A with equalizersand assume that C preserves equalizers. Let D = ( D, ∆ D , ε D) be a comonadover a category B with equalizers and let P, Q : B → A be C-D-bicomodule functors.Let f : P → Q be a functorial morphism of left C-comodule functors and ofright D-comodule functors. Then there exists a unique functorial morphism of leftC-comodule functorsf D : P D → Q Dsatisfyingι Q ◦ f D = ( f D U ) ◦ ι P .Then we can considerC f D : C P D → C Q Dsuch thatD U C f D = f D .□□□