Kerodon

Example 7.1.8.12. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $\operatorname{ev}_0, \operatorname{ev}_1: \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \rightarrow \operatorname{\mathcal{C}}$ be the evaluation functors. Suppose we are given a diagram $\overline{f}: K^{\triangleright } \rightarrow \operatorname{Fun}( \Delta ^1, \operatorname{\mathcal{C}})$. If $\operatorname{ev}_{0} \circ \overline{f}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, then $\overline{f}$ is an $\operatorname{ev}_1$-colimit diagram in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$. This follows by applying Corollary 7.1.8.11 in the special case $B = \Delta ^1$ and $A = \{ 1\} $. In this case, $\overline{f}$ is also a $U$-colimit, where $U = (\operatorname{ev}_0, \operatorname{ev}_1)$ is the restriction map