Corollary 9.5.5.8. Let $\kappa $ be a small regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $\kappa $-compactly generated, and let $\operatorname{\mathcal{C}}_{< \kappa } \subseteq \operatorname{\mathcal{C}}$ denote the full subcategory spanned by the $\kappa $-compact objects. The following conditions are equivalent:
- $(1)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is cocomplete.
- $(2)$
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete.
- $(3)$
The $\infty $-category $\operatorname{\mathcal{C}}_{< \kappa }$ is $\kappa $-cocomplete.