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

48we get that(Cf−1 ) ◦ C ρ Y = C ρ X ◦ f −1 .Lemma 4.18. Let C = ( C, ∆ C , ε C) be a comonad on a category A, let ( P, ρ C P)be aright C-comodule functor where P : A → B and let ( Q, C ρ Q)be a left C-comodulefunctor. Then any equalizer preserved by P C is also preserved by P and any equalizerpreserved by CQ is also preserved by Q.Proof. Consider the following equalizer□Xx Yfg Zin the category A and assume that P C preserves it. Applying to it functors P Cand P we get the following diagrams in BP ε C XP XP CXρ C P XP xP CxP ε C YP YP CYρ C P YP fP gP CfP CgP ε C ZP ZP CZρ C P ZBy assumption, the second row is an equalizer. Assume that there exists a morphismh : H → P Y such that(P f) ◦ h = (P g) ◦ h.Then, by composing with ρ C P Z we get(ρCP Z ) ◦ (P f) ◦ h = ( ρ C P Z ) ◦ (P g) ◦ hand since ρ C Pis a functorial morphism we obtain(P Cf) ◦ ( ρ C P Y ) ◦ h = (P Cg) ◦ ( ρ C P Y ) ◦ h.Since (P CX, P Cx) = Equ B (P Cf, P Cg), there exists a unique morphism k : H →P CX such that(30) (P Cx) ◦ k = ( ρ C P Y ) ◦ h.By composing with P ε C Y we get(P ε C Y ) ◦ (P Cx) ◦ k = ( P ε C Y ) ◦ ( ρ C P Y ) ◦ hand thus(P x) ◦ ( P ε C X ) ◦ k = h.Let l := ( P ε C X ) ◦ k : H → P X. Then we have(P x) ◦ l = (P x) ◦ ( P ε C X ) ◦ k εC = ( P ε C Y ) ◦ (P Cx) ◦ k(30)= ( P ε C Y ) ◦ ( ρ C P Y ) ◦ h = h.Let l ′ : H → P X be another morphism such that(P x) ◦ l ′ = h.