Corollary 7.3.8.2. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\overline{F}: \operatorname{\mathcal{C}}^{\triangleright } \rightarrow \operatorname{\mathcal{D}}$ be a functor of $\infty $-categories. Suppose that $F = \overline{F}|_{\operatorname{\mathcal{C}}}$ is left Kan extended from a full subcategory $\operatorname{\mathcal{C}}^{0} \subseteq \operatorname{\mathcal{C}}$. Then $\overline{F}$ is a colimit diagram if and only if the composite map
\[ (\operatorname{\mathcal{C}}^{0})^{\triangleright } \hookrightarrow \operatorname{\mathcal{C}}^{\triangleright } \xrightarrow { \overline{F} } \operatorname{\mathcal{D}} \]
is a colimit diagram.