Kerodon

$\Newextarrow{\xRightarrow}{5,5}{0x21D2}$ $\newcommand\empty{}$
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$

Corollary 7.1.6.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, and let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram. Assume that, for each vertex $b \in B$, the composite map $K^{\triangleright } \xrightarrow { \overline{f} } \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \xrightarrow { \operatorname{ev}_ b } \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$. Then $\overline{f}$ is a colimit diagram in $\operatorname{Fun}(B,\operatorname{\mathcal{C}})$.

Proof. Apply Corollary 7.1.6.11 in the special case $\operatorname{\mathcal{D}}= \Delta ^0$ (or Corollary 7.1.6.10) in the special case $A = \emptyset $). $\square$