Construction 11.10.2.9. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a locally cartesian fibration of categories which is equipped with a cleavage $(f,Y) \mapsto \widetilde{f}_{Y}$, let $\chi _{U}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be the transport representation of Construction 11.10.2.4, and let $\int ^{\operatorname{\mathcal{C}}} \chi _{U}$ denote the Grothendieck construction on $\chi _{U}$. We define a functor $T: \int ^{\operatorname{\mathcal{C}}} \chi _{U} \rightarrow \operatorname{\mathcal{E}}$ as follows:
Let $(C,X)$ be an object of $\int ^{\operatorname{\mathcal{C}}} \chi _{U}$, so that $C$ is an object of the category $\operatorname{\mathcal{C}}$ and $X$ is an object of the category $\chi _{U}(C) = \operatorname{\mathcal{E}}_{C} \subseteq \operatorname{\mathcal{E}}$. Then $T(C,X) = X$.
Let $(f,u): (C,X) \rightarrow (D,Y)$ be a morphism in the category $\int ^{\operatorname{\mathcal{C}}} \chi _{U}$, so that $f: C \rightarrow D$ is a morphism in the category $\operatorname{\mathcal{C}}$ and $u: X \rightarrow \chi _{U}(f)(Y) = f^{\ast } Y$ is a morphism in the category $\chi _{U}(C) = \operatorname{\mathcal{E}}_{C}$. Then $T(f,u)$ is the composite morphism
\[ T(C,X) = X \xrightarrow {u} f^{\ast }(Y) \xrightarrow { \widetilde{f}_{Y} } Y = T(D,Y) \]in the category $\operatorname{\mathcal{E}}$.