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

(1 C · φ) (γ · 1 Q ) (1 A · σ) = σφ, i.e. σ is an A-linear map. Since ((Q, φ) , σ) is acomonad functor, the 2-cell σ : Q → C · Q satisfies ( )ε C · 1 Q σ = 1Q and (1 C · φ) φ =( )C(X,C)∆C · 1 Q φ. This means that ((Q, φ) , σ) ∈C(X,A)C (X, X) (γ) is an entwined module.By applying the functor Cmd (F (C)) : Cmd (Mnd (C)) −→ Cmd (BIM (C))to the element((Q, φ) , σ) ∈ Cmd ( Mnd (C) ( (X, 1 X ) , ( )) )1 (X,1X ), 1 X , ((X, A) , (C, γ)) we getCmd (F (C)) (((Q, φ) , σ)) ∈ Cmd ( BIM (C) ( (X, 1 X ) , ( )) )1 (X,1X ), 1 X , ((X, A) , (C, γ))which is an element of C(X,C·A) BIM (X, X), i.e. it is a left C · A-comodule with respectto • A .241Appendix A. Gabriel Popescu TheoremNotation A.<strong>1.</strong> Let A be a Grothendieck category, let U be an object of A andlet B = Hom A (U, U). Assume that U is a generator of A i.e. that the functorHom A (U, −) : A → Mod-B is faithful.Lemma A.<strong>2.</strong> In the assumptions and notations of A.1, let X ∈ A and let λ :U (Hom A(U,X)) → X be the codiagonal morphism of the family (f) f∈(HomA (U,X)) . ThenIm (λ) = X.(HomProof. Let J : Ker (λ) → U A(U,X))be the canonical monomorphism and letλ : U (HomA(U,X)) → X be the codiagonal morphism of the family (f) f∈(HomA (U,X))and, for every f ∈ Hom A (U, X) let i f : U → U (HomA(U,X)) the f-th canonicalinjection. Then we have λ ◦ i f = f. Let χ : X → Coker (λ) be the canonicalprojection and let us assume that χ ≠ 0. Then there exists h : U → X such thatχ ◦ h ≠ 0. Then we have0 ≠ χ ◦ h = χ ◦ λ ◦ i h = 0 ◦ i h = 0.Contradiction. Thus Coker (λ) = 0 and hence X = KerCoker (λ) = Im (λ).□Proposition A.3. In the assumptions and notations of A.1, the functor Hom A (U, −) :A → Mod-B is full.Proof. Let ϕ ∈ Hom B (Hom A (U, X) , Hom A (U, Z)). We have to prove that thereexists a morphism g : X → Z such that ϕ = Hom A (U, g). For any subset F ofHom A (U, X) we denote byi F : U (F ) → U (Hom A(U,X))the canonical injection. If F = {f} we will write i f instead of i {f} . Let λ :U (HomA(U,X)) → X be the codiagonal morphism of the family (f) f∈(HomA (U,X))and letµ : U (HomA(U,X)) → Z be the codiagonal morphism of the family (ϕ (f)) f∈(HomA (U,X)) .Then, for every f ∈ Hom A (U, X) we haveλ ◦ i f = f and µ ◦ i f = ϕ (f) .