Kerodon

Remark 4.8.5.9 (Products). Let $n$ be an integer, let $\{ F_ i: \operatorname{\mathcal{C}}_ i \rightarrow \operatorname{\mathcal{D}}_ i \} _{i \in I}$ be a collection of functors of $\infty $-categories, and let $F: \prod _{i \in I} \operatorname{\mathcal{C}}_ i \rightarrow \prod _{i \in I} \operatorname{\mathcal{D}}_ i$ be their product. If each of the functors $F_ i$ is categorically $n$-connective, then $F$ is categorically $n$-connective. The converse holds if each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ is nonempty. See Remark 4.8.2.10.