Remark 7.6.6.23 (Limits of Continuous Functors). Let $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$ be $\infty $-categories, let $\mathbb {K}$ be a collection of simplicial sets, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor which is the levelwise limit of a diagram
If each of the functors $F_{b}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ is $\mathbb {K}$-continuous, then $F$ is also $\mathbb {K}$-continuous. Stated more informally, the collection of $\mathbb {K}$-continuous functors is closed under (levelwise) limits in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. Similarly, the collection of $\mathbb {K}$-cocontinuous functors is closed under (levelwise) colimits in $\operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{\mathcal{D}})$. See Corollary 7.3.8.8.