Lemma 14.24.2. Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $N : \mathcal{A} \to \mathcal{B}$, and $S : \mathcal{B} \to \mathcal{A}$ be functors. Suppose that

the functors $S$ and $N$ are exact,

there is an isomorphism $g : N \circ S \to \text{id}_\mathcal {B}$ to the identity functor of $\mathcal{B}$,

$N$ is faithful, and

$S$ is essentially surjective.

Then $S$ and $N$ are quasi-inverse equivalences of categories.

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