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

7.5. Herd - Coherd - Herd.7.1<strong>1.</strong> Let τ : Q → QP Q be a herd for a regular formal dual structure M =(A, B, P, Q, σ A , σ B , ) where P : A → B, Q : B → A, A : A → A and B : B → Bare functors that preserve equalizers. Then, by Propositions 6.1 and 6.2, we canconstruct comonads C = ( C, ∆ C , ε C) and D = ( D, ∆ D , ε D) and functorial morphismsC ρ Q : Q → CQ and ρ D Q : Q → QD such that (Q, C ρ Q , ρ D Q ) is a C-Dbicomodule) functor (see Theorem 6.5). Let Q as defined in Proposition 7.<strong>1.</strong> Then(Q, D ρ Q , ρ C is a D-C-bicomodule functor. By Theorem 7.5, we construct a coherdQX = ( C, D, Q, Q, δ C , δ D , χ ) where χ := µ B Q ◦(A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦(QP iQ)◦(QqQ). Then we can construct monads A ′ = (A ′ , m A ′, u A ′) and B ′ = (B ′ , m B ′, u B ′)following respectively Proposition 6.25 and Proposition 6.26 as the coequalizers QQC (χQ)◦(QQδC )QQ x′ A ′QQε C DQQ (Qχ)◦(δD QQ)SQ y′ B ′ε D QQThis means that the following holdm A ′ ◦ (x ′ x ′ ) = x ′ ◦ ( χQ ) and u A ′ ◦ ε C = x ′ ◦ δ C(163) m B ′ ◦ (y ′ y ′ ) = y ′ ◦ ( Qχ ) and u B ′ ◦ ε D = y ′ ◦ δ D .Notation 7.1<strong>2.</strong> With notations of Theorem 6.5 and Proposition 7.1, let h : Q → Pbe defined by settingh = B µ P ◦ ( σ B P ) ◦ (P i) ◦ q.The following theorem reformulates Theorem 3.5 in [BV] in our categorical setting.Theorem 7.13. Let M = (A, B, P, Q, σ A , σ B ) be a tame Morita context and letτ : Q → QP Q be a herd for M such that A and B reflect equalizers and coequalizers.We denote by A ′ and B ′ the monads constructed in Claim 7.1<strong>1.</strong> Then1) There are functorial morphisms ν A : A ′ → A and ν B : B ′ → B such that ν Aand ν B are morphisms of monads.2) If the functorial morphism hQ : QQ → P Q, where h = B µ P ◦ ( σ B P ) ◦(P i)◦q,is an isomorphism, then ν A and ν B are isomorphisms.3) If P C = Q ≃ Q ′ = DP then hQ is an isomorphism and hence ν A and ν Bare isomorphisms of monads.Proof. Note that, since A and B reflect equalizers, by Lemma 6.10, we have aregular herd, i.e. the assumptions (A, u A ) = Equ Fun (u A A, Au A ) and (B, u B ) =Equ Fun (u B B, Bu B ) are fulfilled. We will prove only the statement for the monadB ′ , for A ′ the proof is similar.1) Consider the functorial morphismσ B : QQ → B145□