Proposition 7.1.9.4 (Colimits in Oriented 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 $q: K \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ 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 $\operatorname{\mathcal{C}}_1$.
Then $q$ admits a colimit in the $\infty $-category $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. Moreover, an extension $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ 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.
Proof.
Let us identify $q$ with a triple $(q_0, q_1, \alpha )$, where $q_0 = \operatorname{ev}_0 \circ q$ is a diagram in $\operatorname{\mathcal{C}}_0$, $q_1 = \operatorname{ev}_1 \circ q$ is a diagram in $\operatorname{\mathcal{C}}_1$, and $\alpha : F_0 \circ q_0 \rightarrow F_1 \circ q_1$ is a natural transformation between $K$-indexed diagrams in $\operatorname{\mathcal{C}}$. By assumption, we can extend $q_0$ and $q_1$ to colimit diagrams $\overline{q}_0: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0$ and $\overline{q}_1: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_1$, respectively. Moreover, the composition $F_0 \circ \overline{q}_0$ is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}$, so the restriction map
\[ \operatorname{Hom}_{ \operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}) }( F_0 \circ \overline{q}_0, F_1 \circ \overline{q}_1 ) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}( K, \operatorname{\mathcal{C}}) }( F_0 \circ q_0, F_1 \circ q_1 ) \]
is a trivial Kan fibration (Corollary 7.1.6.20). It follows that $\alpha $ can be extended to a natural transformation $\overline{\alpha }: F_0 \circ \overline{q}_0 \rightarrow F_1 \circ \overline{q}_1$ between $K^{\triangleright }$-indexed diagrams in $\operatorname{\mathcal{C}}$. The triple $( \overline{q}_0, \overline{q}_1, \overline{\alpha } )$ can be identified with a diagram $\overline{q}: K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ having the property that $\operatorname{ev}_0 \circ \overline{q}$ and $\operatorname{ev}_1 \circ \overline{q}$ are colimit diagrams in $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$, respectively. It follows from Lemma 7.1.9.3 that any such extension is a colimit diagram in the $\infty $-category $\operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. Conversely, if $\overline{q}': K^{\triangleright } \rightarrow \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is a colimit diagram satisfying $\overline{q}'|_{K} = q$, then $\overline{q}'$ is isomorphic to $\overline{q}$ (as an object of the $\infty $-category $\operatorname{Fun}( K^{\triangleright }, \operatorname{\mathcal{C}}_0 \vec{\times }_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$)), so that $\operatorname{ev}_0 \circ \overline{q}'$ and $\operatorname{ev}_1 \circ \overline{q}'$ are colimit diagrams in $\operatorname{\mathcal{C}}_0$ and $\operatorname{\mathcal{C}}_1$.
$\square$