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

925)σ A = ( ε C A ) ◦ ( Cσ A) ◦ (C ρ Q P )6)σ B = ( Bε D) ◦ ( σ B D ) ◦ ( )P ρ D QFrom the last equalities, we deduce that, σ A is a regular epimorphism if and onlyif so is ε C A and σ B is a regular epimorphism if and only if so is Bε D .Proof. See the dual Theorem 6.30.6.<strong>2.</strong> Herds. Following [BV], we recall some definition about herds.Definition 6.7. A formal dual structure on two categories A and B is a sextupleM = (A, B, P, Q, σ A , σ B ) where A = (A, m A , u A ) and B = (B, m B , u B ) are monadson A and B respectively and ( )A, B, P, Q, σ A , σ B , u A , u B is a preformal dual structure.Moreover ( P : A → B, B µ P : BP → P, µ A P : P A → P ) (andQ : B → A, A µ Q : AQ → Q, µ B Q : QB → Q) are bimodule functors; σ A : QP →A, σ B : P Q → B are subject to the following conditions: σ A is A-bilinear andσ B is B-bilinear(80) σ A ◦ (A µ Q P ) = m A ◦ ( Aσ A) and σ A ◦ ( )Qµ A P = mA ◦ ( σ A A )(81) σ B ◦ (B µ P Q ) = m B ◦ ( Bσ B) and σ B ◦ ( )P µ B Q = mB ◦ ( σ B B )and the associative conditions hold(82)A µ Q ◦ ( σ A Q ) = µ B Q ◦ ( Qσ B) and B µ P ◦ ( σ B P ) = µ A P ◦ ( P σ A) .Definition 6.8. Consider a formal dual structure M = (A, B, P, Q, σ A , σ B ) in thesense of the previous definition. A herd for M is a pretorsor τ : Q → QP Q i.e.(83) (QP τ) ◦ τ = (τP Q) ◦ τ,(84) (σ A Q) ◦ τ = u A Qand(85) (Qσ B ) ◦ τ = Qu B .Definition 6.9. A formal dual structure M = (A, B, P, Q, σ A , σ B ) will be calledregular whenever ( A, B, P, Q, σ A , σ B , u A , u B)is a regular preformal dual structure.In this case a herd for M will be called a regular herd.Lemma 6.10. Let M = (A, B, P, Q, σ A , σ B ) be a formal dual structure and let τ :Q → QP Q be a herd for M. Assume that the underlying functors A and B reflectequalizers. Then τ is a regular herd.Proof. Since A and B are monads, we have m A ◦(Au A ) = Id A and m B ◦(Bu B ) = Id B .Thus, Au A and Bu B are split monomorphisms and thus monomorphisms. Since Aand B reflect equalizers, we deduce that also u A and u B are monomorphisms andthus (A, u A ) = Equ Fun (u A A, Au A ) and (B, u B ) = Equ Fun (u B B, Bu B ), i.e. τ is aregular herd.□□