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.