Example 8.6.8.8. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \mathbf{Cat}$ be a functor of $2$-categories, and let $\int ^{\operatorname{\mathcal{C}}} \mathscr {F}$ be the category of elements of $\mathscr {F}$ (Definition 5.6.1.4). Then the forgetful functor
is a cartesian fibration of $\infty $-categories (Corollary 5.6.1.16). The contravariant transport representation of $U$ can be identified with the Duskin nerve of the functor $\mathscr {F}$, where we identify $\operatorname{N}_{\bullet }^{\operatorname{D}}( \operatorname{Pith}(\mathbf{Cat}) )$ with a full subcategory of $\operatorname{\mathcal{QC}}$ (see Remark 5.5.4.10). This follows by combining Example 8.6.1.9 with Proposition 5.6.3.4.