Remark 7.6.6.6. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. The following conditions are equivalent:
The $\infty $-category $\operatorname{\mathcal{C}}$ is $\kappa $-complete: that is, it admits $K$-indexed limits for every $\kappa $-small simplicial set $K$.
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 7.1.1.15 and Proposition 4.7.5.5. 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).