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

56Notation 4.3<strong>1.</strong> 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 A preserves them. Let Q : B → A be a C-D-bicomodule functor. Inview of Proposition 4.30, we setC Q D = (C Q ) D=C ( Q D) .Proposition 4.3<strong>2.</strong> 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 Bhave equalizers and let Q : B → A be an C-D-bicomodule functor. Then, withnotations of Proposition 4.29, we can consider the functor Q D where ( (Q D , ι Q) =Equ Fun ρD DQ U, Q D Uγ D) . Then(39) Q DD F = Q and ι QD F = ρ D Q.Proof. By construction we have that ( Q D , ι Q) = Equ Fun(ρDQ D U, Q D Uγ D) . By applyingit to the functor D F we get that(Q DD F , ι QD F ) = Equ Fun(ρDQ D U D F , Q D Uγ DD F )= Equ Fun(ρDQ D, Q∆ D) .Since Q is a right D-comodule functor, by Proposition 4.15 we have that(Q, ρDQ)= EquFun(ρDQ D, Q∆ D)so that we get(Q DD F , ι QD F ) (= Equ Fun ρDQ D, Q∆ D) = ( )Q, ρ D Q .□Proposition 4.33. Let D = ( D, ∆ D , ε D) be a comonad over a category B withequalizers such that D preserves equalizers. Let G : D B → A be a functor preservingequalizers. SetQ = G ◦ D F and let ρ D Q = Gγ DD FThen ( Q, ρ D Q)is a right D-comodule functor and(40) Q D = ( G ◦ D F ) D= G.Proof. We compute(ρDQ D ) ◦ ρ D Q = ( Gγ DD F D ) ◦ ( Gγ DD F ) γ D = ( G D F D Uγ DD F ) ◦ ( Gγ DD F )and= ( G D F ∆ D) ◦ ( Gγ DD F ) = ( Q∆ D) ◦ ρ D Q(QεD ) ◦ ρ D Q = ( G D F ε D) ◦ ( Gγ DD F ) adj= G D F = Q.Thus ( Q, ρ D Q)is a right D-comodule functor. Recall that (see Proposition 4.29)(Q D , ι Q) = Equ Fun(ρDQ D U, Q D Uγ D)and by Proposition 4.32 we have Q DD F = Q and ι QD F = ρ D Q . In particular we getQ DD F = Q = G D F .