Kerodon

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

Corollary 7.1.6.5. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $K$ be a simplicial set, and let

\[ U: \operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(K, \operatorname{\mathcal{C}}) \]

denote the restriction map. Then a morphism of simplicial sets $\overline{f}: K^{\triangleleft } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if it is $U$-initial when viewed as an object of the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$.

Proof. Apply Proposition 7.1.6.3 in the special case $\operatorname{\mathcal{D}}= \Delta ^{0}$. $\square$