Warning 7.5.5.4. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories. In general, the condition that $\overline{\mathscr {F}}$ is a categorical limit diagram (in the sense of Definition 7.5.5.1) is independent of the condition that it is a homotopy limit diagram (in the sense of Definition 7.5.4.8): see Exercises 4.5.2.12 and 4.5.2.13.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$