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

)) )Since(Ap QB ̂QA ◦(xQ B U B ̂QA ◦(QP Qpb Q◦ (QP Ql A U) is an epimorphism, weget that ρ ◦ ( A µ ̂Q QB B A ) = ( A Uλ A ) ◦ (Aρ). Therefore ρ is a morphism of A-leftmodule functors and hence, in view of Proposition 3.25, it gives rise to a functorialisomorphism A Q BB ̂QA ∼ = A A.□8.<strong>2.</strong> Equivalence for comodule categories coming from pretorsor. The <strong>results</strong>obtained in this subsection can be found in [BM].Given categories A and B, under the assumptions of Theorem 6.5, one can provethat there exist a comonad C on A and a comonad D on B such that their categoriesof comodules are equivalent. We outline that the assumptions quoted above aresatisfied in the particular case of a regular herd.Using the functors Q and Q, we construct the functors C Q D : D B → C A andD Q C : C A → D B which will be used to set the equivalence between these comodulecategories.Proposition 8.10. In the setting of Theorem 6.5 there exists a functor C ( Q D) :D B → C A such that C U C ( Q D) = Q D where ( Q D , ι Q) = Equ Fun(ρDQ D U, Q D Uγ D) .Moreover we have(204)( Cρ Q D U ) ◦ ι Q = ( Cι Q) ◦ C ρ Q Dwhere C ρ Q D = C Uγ C C ( Q D) : Q D → CQ D .Proof. In view of Theorem 6.5, we can apply Proposition 4.29.8.1<strong>1.</strong> In light of Proposition 8.10, a functor Q : B → A introduced in Theorem 6.5induces a functor C ( Q D) : D B → C A for the comonads C and D. Our next task is toprove that the D-C-bicomodule functor Q, constructed in Proposition 7.1, inducesa functor D (Q C) : C A → D B which yields the inverse of C ( Q D) .Proposition 8.1<strong>2.</strong> Within the assumptions and notations of Theorem 6.5, there )exists a functor D Q C : C A → D B such that D U D Q C = Q C where(Q C , ι Q =(Equ Fun ρ C QC U, Q C Uγ C) . Moreover we have)(205)(Dι Q ◦ D ρ C Q= (D ρ C Q U ) ◦ ι Q( )where D ρ C Q= D Uγ DD Q C : Q C → DQ C so that Q C , D ρ C Qis a left D-comodulefunctor.Proof. In view of Proposition 7.1, we can apply Proposition 4.29 where Q is Q andwe exchange the role of A and B, C and D.□Within the assumptions and notations of Theorem 6.5, one can construct functorialisomorphism D Q C C Q D ∼ = D B and C Q DD Q C ∼ = C A. Such a result can be obtainedby dualizing all the ingredients proved in details for the equivalence between modulecategories.171□