Kerodon

Corollary 4.8.6.21. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories, let $B$ be a simplicial set, and let $n$ be an integer. If $F$ is essentially $n$-categorical, then the induced functor $\operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{D}})$ is also essentially $n$-categorical.