Proposition 9.2.1.7. Let $\kappa $ and $\lambda $ be regular cardinals satisfying $\kappa \trianglelefteq \lambda $. For every $\infty $-category $\operatorname{\mathcal{C}}$, the following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\lambda $-cocomplete.
The $\infty $-category $\operatorname{\mathcal{C}}$ is both $\kappa $-cocomplete and $(\kappa ,\lambda )$-cocomplete.
Proof of Proposition 9.2.1.7. The implication $(1) \Rightarrow (2)$ is trivial (and does not require the assumption $\kappa \trianglelefteq \lambda $). To prove the converse, we proceed as in the proof of Proposition 9.2.1.1. Assume that $\operatorname{\mathcal{C}}$ is both $\kappa $-cocomplete and $(\kappa ,\lambda )$-cocomplete; we wish to show that every $\lambda $-small diagram $K \rightarrow \operatorname{\mathcal{C}}$ admits a colimit. Using Lemma 9.1.7.18, we can realize $K$ as the colimit of a diagram of $\kappa $-small simplicial subsets $\{ K_{\alpha } \} _{\alpha \in A}$ indexed by a $\lambda $-small $\kappa $-directed partially ordered set $(A, \leq )$. Since $\operatorname{\mathcal{C}}$ is $\kappa $-cocomplete, it admits $K_{\alpha }$-indexed colimits for each $\alpha \in A$. Our assumption that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-cocomplete guarantees that it also admits $\operatorname{N}_{\bullet }(A)$-indexed colimits. Applying Corollary 9.1.6.6, we conclude that $\operatorname{\mathcal{C}}$ admits colimits indexed by $K = \varinjlim _{\alpha \in A} K_{\alpha }$, as desired. $\square$