Remark 7.6.7.5. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:

The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, for every simplicial set $K$ which is $\kappa $-small.

The $\infty $-category $\operatorname{\mathcal{C}}$ admits $K$-indexed limits, for every simplicial set $K$ which is essentially $\kappa $-small.

Moreover, in either case, it suffices to consider the case where $K$ is an $\infty $-category. See Remark 7.1.1.17 and Proposition 5.4.5.5. Similarly, a functor $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ preserves $\kappa $-small limits if and only if it preserves $K$-indexed limits, for every simplicial set $K$ which is essentially $\kappa $-small (and it again suffices to consider the case where $K$ is an $\infty $-category).