Theorem 0.1. B-mod category equivalence theorem.
Let 𝒜 be an abelian category with arbitrary direct sums (or coproducts). Also, let P in 𝒜 be
a compact projective generator and set B = (End𝒜P)op. The functor hom
𝒜(P,−−) yields
an equivalence of categories between 𝒜 and the category B − mod.
Proof. The proof proceeds in two steps. At the first step one shows that the functor
is fully faithful, and therefore, at the second step one can apply the Abelian category equivalence
lemma to yield the sought for equivalence of categories.
