PlanetPhysics/B Mod Category Equivalence Theorem

\begin{theorem}{\mathbf B-mod category equivalence theorem.}

Let $$\mathcal{A}$$ be an abelian category with arbitrary direct sums (or coproducts). Also, let $$P$$ in $$\mathcal{A}$$ be a compact projective generator and set $$B = (End_{\mathcal{A}} P)^{op}$$. The functor $$hom_\mathcal{A}(P,--)$$ yields an equivalence of categories between $$\mathcal{A}$$ and the category $$B-mod$$. \end{theorem}

Proof. The proof proceeds in two steps. At the first step one shows that the functor $$F(X) = hom_{\mathcal{A}}(P,X)$$ 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.