Let $k$ be a field. It's well known and easy to prove that a $k$-algebra $R$ may be recovered from the functor $F: R-Mod \to Vect_k$ as the endomorphisms of $F$. Now suppose $\mathcal{C}$ is a category equipped with a functor $F: \mathcal{C} \to Vect_k$. What are the conditions, if any, on $\mathcal{C}$ and $F$ that allow us to conclude that $\mathcal{C} \cong End(F)- Mod$? There is an obvious functor $\mathcal{C} \to End(F)-Mod$, but I'm not sure under what conditions this is essentially surjective.

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necessaryconditions. I don't think they're enough to get the equivalence $\endgroup$ – Max Jun 20 at 21:42