PlanetPhysics/B Mod Category Equivalence Theorem

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

Let $𝒜$ be an [[../AbelianCategory2/|abelian category]] with arbitrary direct sums (or coproducts). Also, let $P$ in $𝒜$ be a compact projective generator and set $B=(En{d}_{𝒜}P{)}^{op}$. The [[../TrivialGroupoid/|functor]] $ho{m}_{𝒜}(P,--)$ yields an equivalence of [[../Cod/|categories]] between $𝒜$ and the category $B-mod$. \end{theorem}

Proof. The proof proceeds in two steps. At the first step one shows that the functor $F(X)=ho{m}_{𝒜}(P,X)$ is [[../FullyFaithfulFunctor2/|fully faithful]], and therefore, at the second step one can apply the [[../AbelianCategoryEquivalenceLemma/|Abelian category equivalence lemma]] to yield the sought for equivalence of categories.