Kerodon

Corollary 4.8.7.21. Let $m$ and $n$ be nonnegative integers, let $B$ be a simplicial set of dimension $\leq m$, and let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be functor of $\infty $-categories which is categorically $(m+n)$-connective. Then the induced map $\operatorname{Fun}(B, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{D}})$ is categorically $n$-connective.