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

129Let σ B : P Q → B be a functorial morphism such that(Qσ B ) ◦ τ = Qu Band let σ A : QP → A be a functorial morphism such that(σ A Q) ◦ τ = u A Q.Then there is a formal codual structure X = (C, D, Q, Q, δ C , δ D ).Proof. In view of Theorem 6.5 and Propositions 7.1, 7.2 a formal codual structureX = (C, D, Q, Q, δ C , δ D ) has been constructed.□Theorem 7.5. Let A and B be categories with equalizers and letM = (A, B, P, Q, σ A , σ B ) be a regular formal dual structure where P : A → B,Q : B → A, A : A → A and B : B → B are functors that preserve equalizers.Let τ : Q → QP Q be a pretorsor. Then there is a formal codual structure X =(C, D, Q, Q, δ C , δ D ). Define χ : QQQ → Q by settingχ := µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) .Then χ is a coherd in X.Proof. By Theorem 7.4 X = (C, D, Q, Q, δ C , δ D ) is a formal codual structure. Toshow that χ is a coherd in X, we have to prove that it satisfies the following conditions.1) Coassociativity, in the sense that χ ◦ ( χQQ ) = χ ◦ ( QQχ ) . Let us computeχ ◦ ( QQχ )= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QqQ) ◦ ( QQχ )q= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP iQ) ◦ (QP Cχ) ◦ ( QqQQQ )i= µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ ( σ A QP Q ) ◦ (QP QP χ) ◦ ( QP iQQQ ) ◦ ( QqQQQ )σ A = µ B Q ◦ (A µ Q B ) ◦ ( AQσ B) ◦ (AQP χ) ◦ ( σ A QP QQQ ) ◦ ( QP iQQQ ) ◦ ( QqQQQ )A µ Q= µBQ ◦ ( Qσ B) ◦ (QP χ) ◦ (A µ Q P QQQ ) ◦ ( σ A QP QQQ ) ◦ ( QP iQQQ ) ◦ ( QqQQQ )(82)= A µ Q ◦ ( σ A Q ) ◦ (QP χ) ◦ (A µ Q P QQQ ) ◦ ( σ A QP QQQ ) ◦ ( QP iQQQ ) ◦ ( QqQQQ )σ A = A µ Q ◦ (Aχ) ◦ ( σ A QQQ ) ◦ (A µ Q P QQQ ) ◦ ( σ A QP QQQ ) ◦ ( QP iQQQ ) ◦ ( QqQQQ ) .We haveA µ Q ◦ (Aχ) ◦ ( σ A QQQ ) ◦ (A µ Q P QQQ )= A µ Q ◦ ( Aµ B Q)◦(A A µ Q B ) ◦ ( AAQσ B) ◦ ( Aσ A QP Q ) ◦ (AQP iQ) ◦ (AQqQ)◦ ( σ A QQQ ) ◦ (A µ Q P QQQ )Qbimod,σ A=A µ Q ◦ ( A A µ Q)◦(AAµBQ)◦(Aσ A QB ) ◦ ( AQP Qσ B) ◦ (AQP iQ) ◦ (AQqQ)◦ ( σ A QQQ ) ◦ (A µ Q P QQQ )A µ Q ass= A µ Q ◦ (m A Q) ◦ ( AAµ B Q)◦(Aσ A QB ) ◦ ( AQP Qσ B) ◦ (AQP iQ) ◦ (AQqQ)