Lemma 4.2.19. A functor is an equivalence of categories if and only if it is both fully faithful and essentially surjective.

**Proof.**
Let $F : \mathcal{A} \to \mathcal{B}$ be essentially surjective and fully faithful. As by convention all categories are small and as $F$ is essentially surjective we can, using the axiom of choice, choose for every $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{B})$ an object $j(X)$ of $\mathcal{A}$ and an isomorphism $i_ X : X \to F(j(X))$. Then we apply Lemma 4.2.18 using that $F$ is fully faithful.
$\square$

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