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

94(ii) the functorial morphism can 2 := (A µ Q D ) ◦ ( Aρ D Q): AQ → QD is anisomorphism.Proof. See the dual Theorem 6.37.6.4. Herds and distributive laws.Proposition 6.14 ([Bo]). Let A and B be categories with equalizers and let τ : Q →QP Q be a regular herd for M = (A, B, P, Q, σ A , σ B ) where the underlying functorsP : A → B, Q : B → A, A : A → A and B : B → B preserve equalizers. LetC = ( C, ∆ C , ε C) and D = ( D, ∆ D , ε D) be the associated comonads constructed inProposition 6.1 and in Proposition 6.<strong>2.</strong> Then1) There exists a mixed distributive law between the comonad C and the monadA, Φ : AC → CA such that(iA) ◦ Φ = φ = ( QP σ A) ◦ (τP ) ◦ (A µ Q P ) ◦ (Ai) .2) There exists an opposite mixed distributive law between the comonad D andthe monad B, Ψ : DB → BD such that(Bj) ◦ Ψ = ψ = ( σ B P Q ) ◦ (P τ) ◦ ( P µ B Q)◦ (jB) .Proof. See the dual Proposition 6.38.6.5. Herds and Galois functors.Lemma 6.15. Let M = ( A, B, P, Q, σ A , σ B) be a formal dual structure where Q :B → A, P : A → B and A = (A, m A , u A ) is a monad on the category A andB = (B, m B , u B ) is a monad on B. Assume that both A and B have coequalizers andthat A, QB preserve them. Then σ A : QP → A induces a morphism σ A A : QP A → A Uin A A and hence there exists a morphism A σ A A : AQP A → Id A A such that(86) AU A σ A A = σ A A.Moreover σ A AAF = σ A : QP AA F = QP → A U A F = A.Proof. Let us consider the following diagram with notations of Proposition 3.30QP A A UAA A Uσ A A A UQµ A P AUQP A Uλ Am AA UA A Uλ A QP A U A A Uσ AA UQp P QPAσ A ′AUλ ASince by assumption QB preserves coequalizers, by Lemma 3.19 also Q preservescoequalizers. Since ( A Uλ A ) ◦ ( σ A AU ) coequalizes the pair ( Qµ A P AU, QP A Uλ A)and(QP A , Qp P ) = Coequ Fun(QµAP AU, QP A Uλ A), by the universal property of the coequalizer,there exists a unique morphism σ A A : QP A → A U such thatσ A A ◦ (Qp P ) = ( A Uλ A ) ◦ ( σ A AU ) .We now want to prove that σ A A : QP A = A U A QP A → A U is a morphism between leftA-module functors which satisfies( A Uλ A ) ◦ ( Aσ A A)= σAA ◦ (A µ Q P A).AU□□