Kerodon

Remark 4.6.4.5. Let $F: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{E}}$ and $G: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{E}}$ be functors of $\infty $-categories. Then we can identify objects of the oriented fiber product $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ with triples $(C,D,e)$, where $C$ is an object of $\operatorname{\mathcal{C}}$, $D$ is an object of $\operatorname{\mathcal{D}}$, and $e: F(C) \rightarrow G(D)$ is a morphism in the $\infty $-category $\operatorname{\mathcal{E}}$. Note that the homotopy fiber product $\operatorname{\mathcal{C}}\times ^{\mathrm{h}}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ of Construction 4.5.2.1 can be identified with the full subcategory of $\operatorname{\mathcal{C}}\operatorname{\vec{\times }}_{\operatorname{\mathcal{E}}} \operatorname{\mathcal{D}}$ spanned by those triples $(C,D,e)$ where the morphism $e$ is an isomorphism.