Corollary 7.5.8.9. Let $\operatorname{\mathcal{C}}$ be a small category and let $\overline{\mathscr {F}}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{QCat}$ be a diagram of $\infty $-categories. Then $\overline{\mathscr {F}}$ is a categorical colimit diagram (in the sense of Definition 7.5.7.3) if and only if the induced functor of $\infty $-categories
\[ \operatorname{N}_{\bullet }^{\operatorname{hc}}( \overline{\mathscr {F}}): \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}^{\triangleright } ) \rightarrow \operatorname{N}_{\bullet }^{\operatorname{hc}}( \operatorname{QCat}) = \operatorname{\mathcal{QC}} \]
is a colimit diagram (in the sense of Variant 7.1.2.5).