Corollary 7.1.7.13. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category, let $B$ be a simplicial set, let $A \subseteq B$ be a simplicial subset, and let $U: \operatorname{Fun}(B,\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}(A,\operatorname{\mathcal{C}})$ be the restriction functor. Let $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}(B,\operatorname{\mathcal{C}})$ be a diagram satisfying the following condition:
- $(\ast )$
Let $\sigma : \Delta ^ n \rightarrow B$ be an $n$-simplex of $B$ which is not contained in $A$ and set $b = \sigma (0)$. Then 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 the $\infty $-category $\operatorname{\mathcal{C}}$.
Then $\overline{f}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{Fun}(B, \operatorname{\mathcal{C}})$.