Corollary 9.5.7.8. An $\infty $-category $\operatorname{\mathcal{C}}$ is presentable if and only if there exists a small $\infty $-category $\operatorname{\mathcal{K}}$, a small collection of morphisms $W$ of the cocompletion $\widehat{\operatorname{\mathcal{K}}} = \operatorname{Fun}( \operatorname{\mathcal{K}}^{\operatorname{op}}, \operatorname{\mathcal{S}})$, and an equivalence of $\operatorname{\mathcal{C}}$ with the full subcategory of $\widehat{\operatorname{\mathcal{K}}}$ spanned by the $W$-local objects.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Using Remark 9.5.6.6, we see that there exists a small $\infty $-category $\operatorname{\mathcal{K}}$ such that $\operatorname{\mathcal{C}}$ is a Bousfield localization of $\widehat{\operatorname{\mathcal{K}}}$. The desired result now follows from the classification of Bousfield localizations supplied by Proposition 9.5.7.5. $\square$