Corollary 7.4.1.19. Let $\kappa $ be an uncountable cardinal and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is is locally $\kappa $-small. For every object $X \in \operatorname{\mathcal{C}}$, the functors
\[ h^{X}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \kappa } \quad \quad h_{X}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{S}}_{< \kappa } \]
preserve $K$-indexed limits, for every simplicial set $K$.