Construction 11.10.5.7 (The Transport Representation). Let $U: \operatorname{\mathcal{D}}\rightarrow \operatorname{\mathcal{C}}$ be a fibration in sets. We define a functor $\chi _{U}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{Set}$ as follows:
For each object $X \in \operatorname{\mathcal{C}}$, we define $\chi _{U}(X)$ to be the set of objects $\operatorname{Ob}( \operatorname{\mathcal{D}}_{X} )$.
For each morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we define $\chi _{U}(f)$ to be the function $f^{\ast }: \operatorname{Ob}( \operatorname{\mathcal{D}}_{Y} ) \rightarrow \operatorname{Ob}( \operatorname{\mathcal{D}}_{X} )$ of Construction 11.10.5.1.
The well-definedness of this functor follows from Example 11.10.5.5 and Proposition 11.10.5.6. We will refer to $\chi _{U}$ as the transport representation associated to $U$.