Kerodon

Variant 11.10.5.2. Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be an opfibration in sets and let $f: X \rightarrow Y$ be a morphism in the category $\operatorname{\mathcal{C}}$. For each object $\widetilde{X} \in \operatorname{\mathcal{D}}_{X}$, there exists a unique pair $(\widetilde{Y}, \widetilde{f} )$, where $\widetilde{Y}$ is an object of the fiber $\operatorname{\mathcal{D}}_{Y}$ and $\widetilde{f}: \widetilde{X} \rightarrow \widetilde{Y}$ satisfies $U( \widetilde{f} ) = f$. Note that the object $\widetilde{Y}$ depends only on $f$ and $\widetilde{X}$. To emphasize this dependence, we will denote the object $\widetilde{Y}$ by $f_{!}( \widetilde{X})$. The construction $\widetilde{X} \mapsto f_!( \widetilde{X} )$ then determines a function $f_!: \operatorname{Ob}( \operatorname{\mathcal{D}}_{X} ) \rightarrow \operatorname{Ob}( \operatorname{\mathcal{D}}_{Y} )$.