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

( ) (= B µb Q◦ m B ̂Q ◦ By ̂Q) (◦ yP Q ̂Q)◦ (P QP Ql) ◦ (P QP QP x)( ) ((162)= B µb Q◦ σ B ̂Q ◦ ν 0Q ′ ̂Q) (◦ yP Q ̂Q)◦ (P QP Ql) ◦ (P QP QP x)(Since ν 0Q ′ ̂Q) (◦ yP Q ̂Q)◦ (P QP Ql) ◦ (P QP QP x) is an epimorphism, we get that( ) ( )B µb Q◦ σ B ̂Q = µ A Q b ◦ ̂QσA. □7.4. From coherds to herds.7.8. Given a coherd χ : QP Q → Q in a formal codual structureX = (C, D, P, Q, δ C , δ D ), our purpose is to build the formal dual structure M =(A, B, ̂Q, Q, σ A , σ B ) and then an herd τ : Q → Q ̂QQ in M.Theorem 7.9. Let A and B be categories with coequalizers and let P : A → B,Q : B → A, C : A → A and D : B → B be functors. Assume that all thefunctors P, Q, C and D preserve coequalizers. Let ε C : C → A and ε D : D → Bbe functorial epimorphisms and assume that ( A, ε C) (= Coequ Fun Cε C , ε C C ) ( ) (andB, εD= Coequ Fun Dε D , ε D D ) . Let χ : QP Q → Q be a functorial morphism suchthatχ ◦ (QP χ) = χ ◦ (χP Q) .Let δ C : C → QP be a functorial morphism such thatχ ◦ (δ C Q) = (ε C Q)and let δ D : D → P Q be a functorial morphism such thatχ ◦ (Qδ D ) = (Qε D ).Then there is a formal dual structure M = (A, B, ̂Q, Q, σ A , σ B ).Proof. In view of Theorem 6.29 and Propositions 7.6, 7.7 a formal dual structureM = (A, B, ̂Q, Q, σ A , σ B ) has been constructed.□Theorem 7.10. Let A and B be categories with coequalizers and letX = (C, D, P, Q, δ C , δ D ) be a regular formal codual structure where P : A → B,Q : B → A, C : A → A and D : B → B are functors that preserve coequalizers.Let χ : QP Q → Q be a copretorsor. Then there is a formal dual structure M =(A, B, ̂Q, Q, σ A , σ B ). Define τ : Q → Q ̂QQ by settingτ := (QlQ) ◦ (QP xQ) ◦ (δ C QP Q) ◦ (CQδ D ) ◦ (C ρ Q D ) ◦ ρ D Q.Then τ is an herd in M.Proof. By Theorem 7.9, M = (A, B, ̂Q, Q, σ A , σ B ) is a formal dual structure. To showthat τ is an herd in M, we have to prove ( that it satisfies the following conditions.1) Associativity, in the sense that Q ̂Qτ) (◦ τ = τ ̂QQ)◦ τ. We have(Q ̂Qτ)◦ τ(= Q ̂Qτ)◦ (QlQ) ◦ (QP xQ) ◦ (δ C QP Q) ◦ (CQδ D ) ◦ (C ρ Q D ) ◦ ρ D Q141