Proposition 9.5.5.3. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\kappa $ be a small regular cardinal. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-presentable.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is an $\operatorname{Ind}_{\kappa }$-completion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$ which is $\kappa $-cocomplete.
Moreover, if these conditions are satisfied, then we can take $\operatorname{\mathcal{C}}_0$ to be the full subcategory $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ spanned by the $\kappa $-compact objects.