Proposition 8.6.4.22. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor of ordinary categories, and let $\mathscr {F}': \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ denote the functor given on objects by $C \mapsto \mathscr {F}(C)^{\operatorname{op}}$. Then the fibrations
\[ \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) \leftarrow \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}) \]
are cocartesian dual to one another.