Lemma 56.2.6. Let $A$ be a ring. Let $\mathcal{B}$ be an additive category with arbitrary direct sums and cokernels. There is an equivalence of categories between

1. the category of functors $F : \text{Mod}_ A \to \mathcal{B}$ which are right exact and commute with arbitrary direct sums, and

2. the category of pairs $(K, \kappa )$ where $K \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ and $\kappa : A \to \text{End}_\mathcal {B}(K)$ is a ring homomorphism

given by the rule sending $F$ to $F(A)$ with its natural $A$-action.

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