Kerodon

Remark 4.5.2.2. 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. It follows from Corollary 4.4.5.5 that the projection map $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1 \rightarrow \operatorname{\mathcal{C}}_0 \times \operatorname{\mathcal{C}}_1$ is an isofibration. In particular, the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ is an $\infty $-category. By construction, the objects of $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ can be identified with triples $(C_0, C_1, e)$, where $C_0$ is an object of $\operatorname{\mathcal{C}}_0$, $C_1$ is an object of $\operatorname{\mathcal{C}}$, and $e: F_0(C_0) \rightarrow F_1(C_1)$ is an isomorphism in the $\infty $-category $\operatorname{\mathcal{C}}$.