Kerodon

Example 11.10.5.5. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a fibration in sets, and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then the function $\operatorname{id}_{X}^{\ast }: \operatorname{Ob}( \operatorname{\mathcal{D}}_{X} ) \rightarrow \operatorname{Ob}( \operatorname{\mathcal{D}}_{X} )$ is the identity map (for each object $\widetilde{X} \in \operatorname{Ob}( \operatorname{\mathcal{D}}_ X )$, the identity morphism $\operatorname{id}_{ \widetilde{X} }$ witnesses the equality $\operatorname{id}_{X}^{\ast }( \widetilde{X} ) = \widetilde{X}$). Similarly, if $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ is an opfibration in sets, then the function $\operatorname{id}_{X!}: \operatorname{Ob}( \operatorname{\mathcal{D}}_{X} ) \rightarrow \operatorname{Ob}( \operatorname{\mathcal{D}}_{X} )$ is the identity map.