$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 7.5.5.8. Let $\operatorname{\mathcal{C}}$ be a category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{QCat}$ be a functor. Then $\overline{\mathscr {F}}$ is a categorical limit diagram if and only if the induced functor of $\infty $-categories
\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \overline{\mathscr {F}}): \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\triangleleft } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]
is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$ (in the sense of Definition 7.1.2.4).
Proof.
By virtue of Corollary 7.5.4.6, the diagram $\operatorname{N}_{\bullet }^{\operatorname{hc}}( \overline{\mathscr {F}} )$ is a limit diagram in the $\infty $-category $\operatorname{\mathcal{QC}}$ if and only if, for every $\infty $-category $\operatorname{\mathcal{E}}$, the diagram of Kan complexes
\[ \operatorname{\mathcal{C}}^{\triangleleft } \rightarrow \operatorname{Kan}\quad \quad C \mapsto \operatorname{Hom}_{ \operatorname{QCat}}( \operatorname{\mathcal{E}}, \overline{\mathscr {F}}(C) )_{\bullet } = \operatorname{Fun}(\operatorname{\mathcal{E}}, \overline{\mathscr {F}}(C) )^{\simeq } \]
is a homotopy limit diagram. Using Proposition 7.5.5.7, we see that this is equivalent to the requirement that $\overline{\mathscr {F}}$ is a categorical limit diagram.
$\square$