Lemma 7.1.9.3 (Colimits in Oriented Fiber Products: Recognition). 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 and let $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ be a diagram satisfying the following conditions:
The composition $(\operatorname{ev}_0 \circ \overline{q}): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0$ is a colimit diagram in $\operatorname{\mathcal{C}}_0$.
The composition $(\operatorname{ev}_1 \circ \overline{q}): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_1$ is a colimit diagram in $\operatorname{\mathcal{C}}_1$.
The composition $(F_0 \circ \operatorname{ev}_0 \circ \overline{q} ): K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}$ is a colimit diagram in $\operatorname{\mathcal{C}}$.
Then $\overline{q}$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$.