$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Lemma 7.1.9.2. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors of $\infty $-categories, let $\operatorname{\mathcal{C}}_0 \vec{\times }_{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ denote the oriented fiber product of Definition 4.6.4.1, and consider the projection maps
\[ \operatorname{\mathcal{C}}_0 \xleftarrow { \operatorname{ev}_0 } \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \xrightarrow { \operatorname{ev}_1 } \operatorname{\mathcal{C}}_1. \]
Suppose we are given a diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ such that $F_0 \circ \operatorname{ev}_0 \circ \overline{q}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$. Then $\overline{q}$ is an $( \operatorname{ev}_0, \operatorname{ev}_1)$-colimit diagram in $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$.
Proof.
By construction, we have a pullback diagram of $\infty $-categories
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \ar [r]^-{G} \ar [d]^{ (\operatorname{ev}_0, \operatorname{ev}_1) } & \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}}) \ar [d]^{U} \\ \operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_1 \ar [r] & \operatorname{Fun}( \operatorname{\partial \Delta }^1, \operatorname{\mathcal{C}}) } \]
where the vertical maps are isofibrations. By virtue of Proposition 7.1.7.6, it will suffice to show that $G \circ \overline{q}$ is a $U$-colimit diagram in the $\infty $-category $\operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$, which is a special case of Example 7.1.8.12.
$\square$