Example 7.4.5.6. Let $F_0: \operatorname{\mathcal{C}}_0 \rightarrow \operatorname{\mathcal{C}}$ and $F_1: \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}$ be functors between small $\infty $-categories, and let $\operatorname{\mathcal{E}}$ denote the oriented fiber product $\operatorname{\mathcal{C}}_0 \operatorname{\vec{\times }}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$. It follows from Corollary 5.3.7.3 that projection onto the second factor determines a cocartesian fibration of $\infty $-categories $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}_{1}$, which admits a covariant transport representation
Now suppose that the functor $F_1$ is a right fibration. In this case, Corollary 6.2.4.8 guarantees that the homotopy fiber product $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}_0 \times _{\operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{C}}_1$ is a coreflective subcategory of $\operatorname{\mathcal{E}}$. Moreover, a morphism in $\operatorname{\mathcal{E}}$ is a $\operatorname{\mathcal{E}}'$-colocal equivalence if and only if it is $U$-cocartesian. It follows that the inclusion functor $\operatorname{\mathcal{E}}' \hookrightarrow \operatorname{\mathcal{E}}$ admits a right adjoint $G: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{E}}'$ (Proposition 6.2.2.18), and that $G$ exhibits $\operatorname{\mathcal{E}}'$ as a localization of $\operatorname{\mathcal{E}}$ with respect to the collection of $U$-cocartesian morphisms (Example 6.3.3.10). Applying Proposition 7.4.5.1, we deduce that $\operatorname{\mathcal{E}}'$ is a colimit of the transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}_1}$ in the $\infty $-category $\operatorname{\mathcal{QC}}$. Stated more informally, we have an equivalence of $\infty $-categories