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

242Let F be a finite subset of Hom A (U, X) and let us consider the commutative diagram0 Ker (λ F )j FU (F ) λ F Xh F0 Ker (λ)i Fj U (Hom A(U,X)) λId X Xwhere λ F : U (F ) → X is the codiagonal morphism of the family (f) f∈F, j and j Fare the canonical inclusions and h F is the morphism that factorizes i F ◦ j F throughKer (λ). We haveµ ◦ i F ◦ j F = µ ◦ j ◦ h F .For every f ∈ F let α f : U → U (F ) and π f : U (F ) → U be respectively the canonicalinjections and projections. ThenId U (F ) = ∑ f∈F α f ◦ π f .Let σ : U → Ker (λ F ) be any morphism. We compute0 = λ F ◦ j F ◦ σ = λ F ◦ Id U (F ) ◦ j F ◦ σ = ∑ f∈F λ F ◦ α f ◦ π f ◦ j F ◦ σ= ∑ f∈F f ◦ π f ◦ j F ◦ σ.Since π f ◦ j F ◦ σ ∈ B = Hom A (U, U) and ϕ ∈ Hom B (Hom A (U, X) , Hom A (U, Z)) ,we get that(∑))0 = ϕ f ◦ π f ◦ j F ◦ σ =f∈F(∑f∈F ϕ (f) ◦ π f ◦ j F ◦ σand hence, since U is a generator of A, we get that∑f∈F ϕ (f) ◦ π f ◦ j F = 0.On the other hand we have thatµ ◦ i F ◦ α f = µ ◦ i f = ϕ (f)and hence we obtain0 = ∑ )µ ◦ i F ◦ α f ◦ π f ◦ j F = µ ◦ i F ◦f∈F(∑f∈F α f ◦ π f ◦ j F= µ ◦ i F ◦ j F = µ ◦ j ◦ h F .Let u : Ker (µ) → U (HomA(U,X)) be the canonical injection.morphism β F : Ker (λ F ) → Ker (µ) such thatj ◦ h F = u ◦ β FThen there exists aand since both j and h F are mono, also β F is mono. We want to check that thefamily (β F ) F ⊆HomA (U,X) is compatible. For every F, G ⊆ Hom A (U, X) finite subsets,let us denote i G F : U (F ) → U (G) . Thus we have λ G ◦ i G F = λ F and0 = λ F ◦ j F = λ G ◦ i G F ◦ j F .