Proposition 9.2.1.14. Let $\kappa $ be a regular cardinal. Then:
If $\kappa $ is uncountable, then an $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\kappa )$-cocomplete if and only if it is idempotent-complete.
If $\kappa = \aleph _0$, then every $\infty $-category is $(\kappa ,\kappa )$-cocomplete.
Proof. We will assume that $\kappa $ is uncountable (the case $\kappa = \aleph _0$ follows from Corollary 9.1.8.12). In this case, the $\infty $-category $\operatorname{N}_{\bullet }(\operatorname{Idem})$ of Construction 8.5.2.7 is both $\kappa $-small and $\kappa $-filtered (Example 9.1.1.10), so every $(\kappa ,\kappa )$-cocomplete $\infty $-category is idempotent-complete (Proposition 8.5.4.10). Conversely, suppose that $\operatorname{\mathcal{C}}$ is idempotent-complete; we wish to show that $\operatorname{\mathcal{C}}$ admits $\operatorname{\mathcal{K}}$-indexed colimits for every $\infty $-category $\operatorname{\mathcal{K}}$ which is both $\kappa $-small and $\kappa $-filtered. Proposition 9.1.8.11 guarantees the existence of a right cofinal functor $\operatorname{N}_{\bullet }(\operatorname{Idem}) \rightarrow \operatorname{\mathcal{K}}$, so the desired result is follows from Corollary 7.2.2.12. $\square$