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

1287. Herds and Coherds7.<strong>1.</strong> Constructing the functor Q. Our next task is to construct a D-C-bicomodulefunctor Q. Such a functor appears in [BM, Section 5], but we give here new notations.For the details of the proofs, see the dual <strong>results</strong> in the following.Proposition 7.<strong>1.</strong> In the setting of Theorem 6.5, we define functors Q : A → B viathe equalizerQ q P C (θl P )◦(P i)(θ r P )◦(P i) BP QPThen there exists a unique functorial morphism κ ′ 0 : Q → DP such that(144) (P i) ◦ q = (jP ) ◦ κ ′ 0Moreover(Q,κ′0)= EquFun((P ωl ) ◦ (jP ) , (P ω r ) ◦ (jP ) ) .( The functor)Q can be equipped with the structure of a D-C-bicomodule functorQ, D ρ Q , ρ C where ρ C and D ρQQ Q are uniquely determined by(145) (qC) ◦ ρ C Q = ( P ∆ C) ◦ qand(146) (Dκ ′ 0) ◦ D ρ Q = ( ∆ D P ) ◦ κ ′ 0.Proposition 7.<strong>2.</strong> In the setting of Theorem 6.5 and Proposition 7.1, there existtwo functorial morphisms δ C : C → QQ and δ D : D → QQ where δ C is C-bicolinearand δ D is D-bicolinear and they fulfill(147) (Qq) ◦ δ C = (iC) ◦ ∆ Cand(148) (κ ′ 0Q) ◦ δ D = (Dj) ◦ ∆ D .Moreover the coassociative conditions hold, that is(δ C Q) ◦ C ρ Q = (Qδ D ) ◦ ρ D Q and ( δ D Q ) ◦ D ρ Q = ( Qδ C)◦ ρCQ .7.<strong>2.</strong> From herds to coherds.7.3. Given an herd τ : Q → QP Q in a formal dual structure M = (A, B, P, Q, σ A , σ B ),our purpose is to build the formal codual structure X = (C, D, Q, Q, δ C , δ D ) and thena coherd χ : QQQ → Q in X.Theorem 7.4. Let A and B be categories with equalizers and let P : A → B,Q : B → A, A : A → A and B : B → B be functors. Assume that all thefunctors P, Q, A and B preserve equalizers. Let u A : A → A and u B : B → Bbe functorial monomorphisms and assume that (A, u A ) = Equ Fun (u A A, Au A ) and(B, u B ) = Equ Fun (u B B, Bu B ).Let τ : Q → QP Q be a functorial morphism such that(QP τ) ◦ τ = (τP Q) ◦ τ