Example 11.3.0.8. Let $q: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{D}}$ be a cartesian fibration of categories (Definition 5.0.0.3), so that the induced map $\operatorname{N}_{\bullet }(q): \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{D}})$ is a cartesian fibration of $\infty $-categories (Example 5.1.4.2). Then the homotopy transport representation $\operatorname{hTr}_{\operatorname{N}_{\bullet }(q)}: \operatorname{\mathcal{D}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{Cat}_{\infty }}$ is given by the composition
Here $\chi _{q}$ denotes the transport representation of Construction 11.10.2.4 (with respect to any cleavage of the fibration $q$), the second functor is the truncation map of Remark 11.6.0.71, and $\operatorname{N}_{\bullet }$ is the fully faithful functor of Remark 4.5.1.3. Stated more informally, the homotopy transport representation $\operatorname{hTr}_{ \operatorname{N}_{\bullet }(q)}$ of Construction 5.2.5.7 can be obtained from the transport representation $\chi _{ \operatorname{N}_{\bullet }(q)}$ of Construction 11.10.2.4 by passing from the $2$-category $\mathbf{Cat}$ to its homotopy category.