Corollary 7.1.9.7 (Colimits in Homotopy Fiber Products: Existence). 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 $q: K \rightarrow \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ be a diagram satisfying the following conditions:
The diagram $(\operatorname{ev}_0 \circ q): K \rightarrow \operatorname{\mathcal{C}}_0$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0$ which is preserved by the functor $F_0$.
The diagram $(\operatorname{ev}_1 \circ q): K \rightarrow \operatorname{\mathcal{C}}_1$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_1$ which is preserved by the functor $F_1$.
Then $q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. Moreover, an extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a colimit diagram if and only if $\operatorname{ev}_0 \circ \overline{q}$ and $\operatorname{ev}_1 \circ \overline{q}$ are colimit diagrams in the $\infty $-categories $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively.