Example 7.1.3.11. Let $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$ be a collection of $\infty $-categories having product $\operatorname{\mathcal{C}}= \prod _{i \in I} \operatorname{\mathcal{C}}_ i$. Then a morphism of simplicial sets $\overline{u}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a limit diagram if and only if, for every index $i \in I$, the composition
is a limit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_ i$. See Example 4.7.1.11.
In particular, if each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ admits $K$-indexed limits, then $\operatorname{\mathcal{C}}$ admits $K$-indexed limits. The converse holds provided that each of the $\infty $-categories $\operatorname{\mathcal{C}}_{i}$ is nonempty.