Kerodon

Example 4.5.2.3. 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. If $\operatorname{\mathcal{C}}$ is a Kan complex, then every morphism in $\operatorname{\mathcal{C}}$ is an isomorphism (Proposition 1.4.6.10): that is, we have $\operatorname{Isom}(\operatorname{\mathcal{C}}) = \operatorname{Fun}(\Delta ^1, \operatorname{\mathcal{C}})$. It follows that the homotopy fiber product $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ of Construction 4.5.2.1 coincides with the homotopy fiber product introduced in Construction 3.4.0.3.