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

102(b) A comonad C = ( C, ∆ C , ε C) on the category A such that the functor Cpreserves equalizers and a mixed distributive law Φ : AC → CA such thatAQ is a Galois comodule functor over ˜C (where ˜C is the lifting of C)(c) A comonad D = ( D, ∆ D , ε D) on the category B such that the functor Dpreserves equalizers and an opposite mixed distributive law Ψ : DB → BDsuch that B P is a Galois comodule functor over ˜D (where ˜D is the lifting ofD).Proof. By Proposition 6.11 the pairs ( A Q, P A ) and ( B P , Q B ) are adjunctions andhence P A and Q B preserve equalizers. Since A = A U A F and B = B U B F preserveequalizers, by Lemma 3.22 also A F and B F preserve them so that, in view of (15),we get that P = P AA F and Q = Q BB F preserve equalizers.(a) ⇒ (b) Assume that τ : Q → QP Q is a herd for M = ( A, B, P, Q, σ A , σ B) . ByProposition 6.14 there exists a mixed distributive law Φ : AC → CA such that(iA) ◦ Φ = ( QP σ A) ◦ (τP ) ◦ (A µ Q P ) ◦ (Ai) .Then, by Theorem 5.7, there exists a lifting comonad ˜C =(˜C, ∆e C, ε e C)on the categoryA A. By Proposition 6.16, there(exists)a functorial morphism eC ρ A Q : A Q → ˜C A Qsuch that A U eC ρ A Q = C ρ Q and AQ, eC ρ A Q is a left ˜C-comodule functor. Since byassumption we have a regular formal dual structure, by Theorem 6.6, the functorialmorphism can 1 := ( Cσ A) ◦ (C ρ Q P ) : QP → CA is an isomorphism and so, byLemma 6.23, A Q is a left ˜C-Galois functor.(b) ⇒ (a) Follows by [BM, Theorem 4.4 (1)] where(T , (N A , R A ), (N B , R B ), C, ξ) = ( A A, ( A F , A U), ( A Q, P A ), C, A U A can A ) noting that apretorsor for a formal dual structure is a herd.□6.7. Copretorsors.Proposition 6.25. Let A and B be categories with coequalizers and let P : A →B, Q : B → A, and C : A → A be functors. Assume that all the functors P, Q andC preserve coequalizers. Let ε C : C → A be a functorial morphism and assume that(A, εC ) = Coequ Fun(Cε C , ε C C ) . Let χ : QP Q → Q be a functorial morphism suchthat(98) χ ◦ (QP χ) = χ ◦ (χP Q)and let δ C : C → QP be a functorial morphism such that(99) χ ◦ (δ C Q) = ε C Q.Let w l = (χP ) ◦ (QP δ C ) and w r = QP ε C : QP C → QP . Set(100) (A, x) = Coequ Fun(w l , w r) .There exists a functorial morphism A µ Q : AQ → Q such that(101)A µ Q ◦ (xQ) = χ.There exist functorial morphisms m A : AA → A and u A : A → A such that A =(A, m A , u A ) is a monad over A that preserves coequalizers. Moreover m A and u A