Proposition 8.6.4.19. Let $\operatorname{\mathcal{C}}$ be a category, let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories, and let $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be the functor given on objects by $C \mapsto \mathscr {F}(C)^{\operatorname{op}}$. Then (the nerves of) the fibrations
\[ \int _{\operatorname{\mathcal{C}}} \mathscr {F} \rightarrow \operatorname{\mathcal{C}}\quad \quad \int _{\operatorname{\mathcal{C}}} \mathscr {F}' \rightarrow \operatorname{\mathcal{C}} \]
are cocartesian dual to one another.