Proposition 9.2.8.3. Let $\operatorname{\mathcal{C}}$ be an essentially finite $\infty $-category and let $W$ be a finite collection of morphisms of $\operatorname{\mathcal{C}}$. Then the localization $\operatorname{\mathcal{C}}[W^{-1}]$ is also essentially finite.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Choose a categorical equivalence $F: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}_0$ is a finite simplicial set. By virtue of Lemma 9.2.8.2, we can assume that $W = F(W_0)$, where $W_0$ is a collection of morphisms of $\operatorname{\mathcal{C}}_0$. In this case, we can identify $\operatorname{\mathcal{C}}[W^{-1}]$ with a localization of $\operatorname{\mathcal{C}}_0$ with respect to $W_0$, which is essentially finite by virtue of Remark 6.3.2.10. $\square$