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

112Then ν : Y → D is the unique morphism satisfying condition (124) so that(D, δ D ) = Equ Fun(ζ ◦ (Dy) ◦( Dρ P Q ) , θ ◦ (Dy) ◦ (D ρ P Q ))i.e. also δ D is a regular monomorphism.6.8. Coherds. Following [BV], by formally dualizing definitions of formal dualstructure and herd, the notions of formal codual structure and of coherd are introduced.Definition 6.3<strong>1.</strong> A formal codual structure on two categories A and B is a sextupleX = (C, D, P, Q, δ C , δ D ) where C = ( C, ∆ C , ε C) and D = ( D, ∆ D , ε D) arecomonads on on A and B respectively and ( C, D, P, Q, δ C , δ D , ε C , ε D) is a preformalcodual structure. Moreover ( (P : A → B, D ρ P : P → DP, ρ C P : P → P C) andQ : B → A, C ρ Q : Q → CQ, ρ D Q : Q → QD) are bicomodule functors; δ C : C →QP, δ D : D → P Q are subject to the following conditions: δ C is C-bicolinear andδ D is D-bicolinear(125) ( C ρ Q P ) ◦ δ C = (Cδ C ) ◦ ∆ C and ( )Qρ C P ◦ δC = (δ C C) ◦ ∆ C( )(126) P ρDQ ◦ δD = (δ D D) ◦ ∆ D and (D ρ P Q ) ◦ δ D = (Dδ D ) ◦ ∆ Dand the coassociative conditions hold(127) (δ C Q) ◦ C ρ Q = (Qδ D ) ◦ ρ D Q and (δ D P ) ◦ D ρ P = (P δ C ) ◦ ρ C P .Definition 6.3<strong>2.</strong> Consider a formal codual structure X = (C, D, P, Q, δ C , δ D ) inthe sense of the previous definition. A coherd for X is a copretorsor χ : QP Q → Qi.e.(128) χ ◦ (χP Q) = χ ◦ (QP χ)(129) χ ◦ (δ C Q) = ε C Qand(130) χ ◦ (Qδ D ) = Qε D .Definition 6.33. A formal codual structure X = (C, D, P, Q, δ C , δ D ) will be calledregular whenever ( C, D, P, Q, δ C , δ D , ε C , ε D) is a regular preformal codual structure.In this case a coherd for X will be called a regular coherd.Lemma 6.34. Let X = (C, D, P, Q, δ C , δ D ) be a formal codual structure and let χ :QP Q → QQ be a coherd for X. Assume that the underlying functors C and D reflectcoequalizers. Then χ is a regular coherd.Proof. Since C and D are comonads, we have ( Cε C) ◦ ∆ C = Id C and ( Dε D) ◦ ∆ D =Id D . Thus, Cε C and Dε D are split epimorphisms and thus epimorphisms. Since Cand D reflect coequalizers, we deduce that also ε C and ε D are epimorphisms andthus ( A, ε C) (= Coequ Fun ε C C, Cε C) and ( B, ε D) (= Coequ Fun ε D D, Dε D) , i.e. χ isa regular coherd.□Proposition 6.35. Let X = (C, D, P, Q, δ C , δ D ) be a formal codual structure suchthat the lifted functors C Q D : D B → C A and D P C : C A → D B determine an equivalenceof categories. Then ( Q D , D P ) and ( P C , C Q ) are adjunctions.□