Construction 11.10.2.4 (The Transport Representation). Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of categories which is equipped with a cleavage $(f,Y) \mapsto \widetilde{f}_{Y}$. We define a lax functor of $2$-categories $\chi _ U: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ as follows:
To each object $C \in \operatorname{\mathcal{C}}$, the functor $\chi _ U$ assigns the fiber $\chi _ U(C) = \operatorname{\mathcal{E}}_{C}$.
To each morphism $f: B \rightarrow C$ in $\operatorname{\mathcal{C}}$, the functor $\chi _ U$ assigns the contravariant transport functor $\chi _ U(f) = f^{\ast }: \operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}_{B}$ of Construction 5.2.2.2.
For each object $C \in \operatorname{\mathcal{C}}$, the unit constraint $\operatorname{id}_{ \chi _ U(C) } \rightarrow \chi _ U( \operatorname{id}_{C} )$ is the isomorphism $\epsilon _{C}: \operatorname{id}_{ \operatorname{\mathcal{E}}_{C} } \simeq \operatorname{id}_{C}^{\ast }$ of Example 11.10.2.1.
For every pair of composable morphisms $f: B \rightarrow C$ and $g: C \rightarrow D$ in $\operatorname{\mathcal{C}}$, the composition constraint $\chi _ U(f) \circ \chi _{U}(g) \rightarrow \chi _{U}(g \circ f)$ is the natural transformation $\mu _{f,g}: f^{\ast } \circ g^{\ast } \rightarrow (g \circ f)^{\ast }$ of Proposition 11.10.2.3.
We will refer to $\chi _{U}$ as the transport representation of the locally cartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ (and the cleavage $(f,Y) \mapsto \widetilde{f}_{Y}$).