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

ξ ↦→ f ◦ ξ.Then it is well-defined a functor Hom A (P, −) : A → Mod-B. We want to prove thatHom A (P, −) is an equivalence of category. To be full and faithful for Hom A (P, −)means that the mapφ A,A ′ : Hom A (A, A ′ ) −→ Hom Mod-B (Hom A (P, A) , Hom A (P, A ′ ))( )HomA (P, f) : Homf ↦→A (P, A) → Hom A (P, A ′ )ξ ↦→ f ◦ ξis bijective for every A, A ′ ∈ A. Note that Hom A (P, −) induces an isomorphismφ P,P : Hom A (P, P ) −→ Hom Mod-B (Hom A (P, P ) , Hom A (P, P )) .In fact, for every ζ ∈ Hom A (P, P ) such that Hom A (P, ζ) = 0 we have that, forevery ξ ∈ Hom A (P, P ) , 0 = Hom A (P, ζ) (ξ) = ζ ◦ ξ. Since P is a generator, wededuce that ζ = 0. Now, let f : Hom A (P, P ) → Hom A (P, P ) be a morphism inMod-B and set f (Id P ) = χ. Then, for every ξ ∈ Hom A (P, P ), we haveand thusf (ξ) = f (Id P ◦ ξ) f∈Mod-B= f (Id P ) ◦ ξ = χ ◦ ξ = Hom A (P, χ) (ξ)f = Hom A (P, χ) = Hom A (P, −) (χ)so that φ P,P is an epimorphism. Let us consider families (P i ) i∈Iand (P j ) j∈JwhereP i ≃ P ≃ P j for every i ∈ I and j ∈ J. Set B i = Hom A (P, P i ) and B j =Hom A (P, P j ). Then B i = Hom A (P, P i ) ≃ Hom A (P, P ) = B and similarly B j ≃ B.Let us compute( ∐Hom A P j , ∐ )coprodP i ≃∏ (Hom A P j , ∐ )P finiteP i ≃∏ ∐Hom A (P j , P i )j∈J i∈Ij∈Ji∈Ij∈J i∈Iφ P,P≃ ∏ ∐Hom Mod-B (Hom A (P, P j ) , Hom A (P, P i )) = ∏ ∐Hom Mod-B (B j , B i )j∈J i∈I j∈J i∈I(B i ≃Bfinite∏≃ Hom Mod-B B j , ∐ )(coprod∐B i ≃ Hom Mod-B B j , ∐ )B ij∈Ji∈Ij∈J i∈Ihence(275) Hom A( ∐j∈JP j , ∐ i∈IP i)≃ Hom Mod-B( ∐j∈JB j , ∐ i∈IB i)which says that Hom A (P, −) induces a bijection between the full subcategory of thecoproducts of copies of P in A and the full subcategory of coproducts of copies ofB in Mod-B, i.e. Hom A (P, −) is full and faithful on the full subcategory of thecoproducts of copies of P in A. Let us denote by ε P i : P → P (I) , p P i : P (I) → P andε j : B = Hom A (P, P ) → B (I) = Hom A (P, P ) (I) , p j : B (I) = Hom A (P, P ) (I) → B =Hom A (P, P ) the canonical maps. Now, let A, A ′ ∈ A. Since P is a generator, wehaveP (J)e f−→ P (I)h−→ A = Coker (h) → 0261