Remark 4.7.9.18. Let $n$ be an integer. Then, for every collection of $\infty $-categories $\{ \operatorname{\mathcal{C}}_ i \} _{i \in I}$, the canonical map
\[ \operatorname {h}_{\mathit{\leq {}n}}{\mathit{( \prod _{i \in I} \operatorname{\mathcal{C}}_ i )}} \rightarrow \prod _{i \in I } \operatorname {h}_{\mathit{\leq {}n}}{\mathit{(\operatorname{\mathcal{C}}_ i)}} \]
is an isomorphism. This follows by inspecting the explicit descriptions supplied by Construction 4.7.9.9 (for the case $n > 0$), Exercise 4.7.9.4 (for the case $n=0$) and Example 4.7.9.5 (for the case $n < 0$).