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

Proposition 6.1<strong>1.</strong> Let M = (A, B, P, Q, σ A , σ B ) be a formal dual structure suchthat the lifted functors A Q B : B B → A A and B P A : A A → B B determine an equivalenceof categories. Then ( A Q, P A ) and ( B P , Q B ) are adjunctions.Proof. Since ( A F , A U) and ( B F , B U) are adjunctions, ( A Q BB F , B U B P A ) = ( A Q, P A )and ( B P AA F , A U A Q B ) = ( B P , Q B ) are also adjunctions.□6.3. Herds and comonads.Theorem 6.12 ([Bo]). Let A and B be categories in both of which the equalizer ofany pair of parallel morphisms exists. Let M = (A, B, P, Q, σ A , σ B ) be a formal dualstructure on two categories A and B. Then we have(1) If C = ( C, ∆ C , ε C) is a comonad on the category A and ( Q, C ρ Q : Q → CQ )is a left C-comodule functor such that(i) the functorial morphism can 1 := ( Cσ A) ◦ (C ρ Q P ) : QP → CA is anisomorphism(ii) the functorial morphism can 2 := ( (Cµ Q) B ◦ Cρ Q B ) : QB → CQ is anisomorphismthen τ := ( can −11 Q ) ◦ (Cu A Q) ◦ C ρ Q : Q → QP Q is a pretorsor and thusa herd.(2) If D = ( D, ∆ D , ε D) is a comonad on the category B and ( Q, ρ D Q : Q → QD)is a right D-comodule functor such that(i) the functorial morphism can 1 := ( σ B D ) ◦ ( P ρQ) D : P Q → BD is anisomorphism(ii) the functorial morphism can 2 := (A µ Q D ) ◦ ( AρQ) D : AQ → QD is anisomorphismthen τ := (Qcan −1 1 ) ◦ (Qu B D) ◦ ρ D Q : Q → QP Q is a pretorsor and thus aherd.Proof. See the dual Theorem 6.36.Theorem 6.13 ([Bo]). Let A and B be categories in both of which the equalizerof any pair of parallel morphisms exists. Let M = (A, B, P, Q, σ A , σ B ) be a regularformal dual structure such that the underlying functors A, B, P and Q preserveequalizers, then the existence of the following structures are equivalent:(a) A herd τ : Q → QP Q in M;(b) A comonad C = ( C, ∆ C , ε C) on the category A such that the functor Cpreserves equalizers and ( Q, C ρ Q : Q → CQ ) is a left C-comodule functorsubject to the following conditions(i) the functorial morphism can 1 := ( Cσ A) ◦ (C ρ Q P ) : QP → CA is anisomorphism(ii) the functorial morphism can 2 := ( Cµ B Q)◦( Cρ Q B ) : QB → CQ is anisomorphism;(c) A comonad D = ( D, ∆ D , ε D) on the category B such that the functor Dpreserves equalizers and ( Q, ρ D Q : Q → QD) is a right D-comodule functorsubject to the following conditions(i) the functorial morphism can 1 := ( σ B D ) ◦ ( P ρ D Q): P Q → BD is anisomorphism93□