An extension of a theorem on endomorphism rings and equivalences (Q1913991)

It is proved that if \({\mathcal C}\) is any Grothendieck category, \(M\) an object of \({\mathcal C}\) and \(S\) the endomorphism ring of \(M\), then the functor \(\Hom_{\mathcal C} (M,-)\) from \({\mathcal C}\) to \(\text{Mod-}S\) establishes an equivalence between \({\mathcal C}_M\) and the quotient category \(\text{Mod-}(S, {\mathcal F})\).