Corollary 7.1.7.6. 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 $\overline{f}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram if and only if it is $U$-initial when viewed as an object of the $\infty $-category $\operatorname{Fun}(K^{\triangleright }, \operatorname{\mathcal{C}})$.