$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$

Corollary Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $f: K \rightarrow \operatorname{Fun}(B, \operatorname{\mathcal{C}})$ be a diagram. Assume that, for every vertex $b \in B$, the diagram

\[ K \xrightarrow {f} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}} \]

has a colimit in $\operatorname{\mathcal{C}}$. Then $f$ can be extended to a morphism $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ having the property that each composition $K^{\triangleright } \xrightarrow {\overline{f}} \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.

Proof. Apply Corollary in the special case $\operatorname{\mathcal{D}}= \Delta ^0$. $\square$