Example 4.5.2.4. 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 ordinary categories. Then the homotopy fiber product $\operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_0) \times ^{\mathrm{h}}_{ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}_1)$ can be identified with the nerve of a category $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$, which can be described concretely as follows:
The objects of $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ are 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 $\operatorname{\mathcal{C}}$.
A morphism from $(C_0, C_1, e)$ to $(C'_0, C'_1, e')$ is a pair $(f_0, f_1)$, where $f_0: C_0 \rightarrow C'_0$ is a morphism in the category $\operatorname{\mathcal{C}}_0$, $f_1: C_1 \rightarrow C'_1$ is a morphism in the category $\operatorname{\mathcal{C}}_1$, and the diagram
\[ \xymatrix { C_0 \ar [r]^{f_0} \ar [d]^{e}_{\sim } & C'_0 \ar [d]^{e'}_{\sim } \\ C_1 \ar [r]^{ f_1 } & C'_1 } \]commutes in the category $\operatorname{\mathcal{C}}$.
We will refer to $\operatorname{\mathcal{C}}_0 \times ^{\mathrm{h}}_{\operatorname{\mathcal{C}}} \operatorname{\mathcal{C}}_1$ as the homotopy fiber product of $\operatorname{\mathcal{C}}_0$ with $\operatorname{\mathcal{C}}_1$ over $\operatorname{\mathcal{C}}$.