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

90Let θ l = ( σ B P Q ) ◦ (P τ) and θ r = u B P Q : P Q → BP Q. Set(64) (D, j) = Equ Fun(θ l , θ r) .There exists a functorial morphism ρ D Q : Q → QD such that(65) (Qj) ◦ ρ D Q = τ.There exist functorial morphisms ∆ D : D → DD and ε D : D → B such thatD = ( D, ∆ D , ε D) is a comonad over B and D preserves equalizers. The functorialmorphisms ∆ D and ε D are uniquely determined by(66) (jD) ◦ ∆ D = ( )P ρ D Q ◦ j and σ B ◦ j = u B ◦ ε Dor equivalently(67) (P τ) ◦ j = (jj) ◦ ∆ D and σ B ◦ j = u B ◦ ε D .Moreover ( Q, ρ D Q)is a right D-comodule functor.Definition 6.3. Let A and B be categories. A preformal dual structure is a eightupleΞ = ( A, B, P, Q, σ A , σ B , u A , u B)where A : A → A, B : B → B, P : A → B andQ : B → A are functors, σ A : QP → A, σ B : P Q → B, u A : A → A, u B : B → B arefunctorial morphisms. A pretorsor τ for Ξ is a functorial morphism τ : Q → QP Qsatisfying the following conditions.1) Associativity, in the sense that(68) (QP τ) ◦ τ = (τP Q) ◦ τ2) Unitality, in the sense that(69) (σ A Q) ◦ τ = u A Qand(70) (Qσ B ) ◦ τ = Qu B .Definition 6.4. A preformal dual structure Ξ = ( P, Q, A, B, σ A , σ B , u A , u B , ) willbe called regular whenever (A, u A ) = Equ Fun (u A A, Au A ) and(B, u B ) = Equ Fun (u B B, Bu B ). In this case a pretorsor for Ξ will be called a regularpretorsor.Theorem 6.5 ([BM, Lemma 4.8]). Let A and B be categories with equalizers and letτ : Q → QP Q be a regular pretorsor for Ξ = ( A, B, P, Q, σ A , σ B , u A , u B). Assumethat the underlying functors P, Q, A and B preserve equalizers. Let ω l = ( QP σ A) ◦(τP ) and ω r = QP u A : QP → QP A. Set(71) (C, i) = Equ Fun(ω l , ω r) .Then there exists a functorial morphism C ρ Q : Q → CQ such that(72) (iQ) ◦ C ρ Q = τ.There exist functorial morphisms ∆ C : C → CC and ε C : C → A such thatC = ( C, ∆ C , ε C) is a comonad over A and C preserves equalizers. The functorialmorphisms ∆ C and ε C are uniquely determined by(73) (τP ) ◦ i = (ii) ◦ ∆ C and σ A ◦ i = u A ◦ ε C .