Remark 7.5.5.5. Let $\operatorname{\mathcal{C}}$ be a category, let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a categorical limit diagram of $\infty $-categories, and define $\overline{\mathscr {F}}^{\simeq }: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}$ by the formula $\overline{\mathscr {F}}^{\simeq }(C) = \overline{\mathscr {F}}(C)^{\simeq }$. Then $\overline{\mathscr {F}}^{\simeq }$ is a homotopy limit diagram. This follows by combining Example 7.5.2.8 with Remark 4.5.1.20.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$