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

Corollary Let $K$ be a simplicial set and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which admits $K$-indexed colimits. Then, for every simplicial set $B$, the $\infty $-category $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$ also admits $K$-indexed colimits. Moreover, a morphism of simplicial sets $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ is a colimit diagram if and only if, for every vertex $b \in B$, the morphism

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

is a colimit diagram in $\operatorname{\mathcal{C}}$.