$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 8.6.3.23. Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be a functor of ordinary categories and let $\mathscr {F}', \operatorname{Tw}(\mathscr {F}): \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ be the functors given on objects by the formulae $\mathscr {F}'(C) = \mathscr {F}(C)^{\operatorname{op}}$ and $\operatorname{Tw}(\mathscr {F})(C) = \operatorname{Tw}( \mathscr {F}(C) )$. Then the tautological map
\[ \lambda = (\lambda _{-},\lambda _{+} ): \operatorname{N}_{\bullet }^{\operatorname{Tw}(\mathscr {F})}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }^{\mathscr {F}^{\operatorname{op}}}(\operatorname{\mathcal{C}}) \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \]
exhibits the fibration $U^{\vee }: \operatorname{N}_{\bullet }^{\mathscr {F}'}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ as a cocartesian dual of the fibration $U: \operatorname{N}_{\bullet }^{\mathscr {F}}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$.
Proof.
For each object $C \in \operatorname{\mathcal{C}}$, Proposition 8.1.1.11 guarantees that the morphism
\[ \lambda _{C}: \operatorname{Tw}( \mathscr {F}(C) ) \rightarrow \mathscr {F}(C)^{\operatorname{op}} \times \mathscr {F}(C) \]
is a left fibration of $\infty $-categories, which is a balanced coupling by virtue of Example 8.2.6.2. Applying Corollary 5.3.3.18, we deduce that $\lambda $ is a left fibration of $\infty $-categories. Let $U: \operatorname{N}_{\bullet }^{ \operatorname{Tw}(\mathscr {F}) }(\operatorname{\mathcal{C}}) \rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ denote the projection map, and let $f: X \rightarrow Y$ be a morphism in the $\infty $-category $\operatorname{N}_{\bullet }^{ \operatorname{Tw}(\mathscr {F}) }(\operatorname{\mathcal{C}})$ having image $u: C \rightarrow D$ in $\operatorname{\mathcal{C}}$. To complete the proof, it will suffice to show that if $X$ is universal for the coupling $\lambda _{C}$ and $f$ is $U$-cocartesian, then $Y$ is universal for the coupling $\lambda _{D}$. Our assumption that $f$ is $U$-cocartesian guarantees that $Y$ is isomorphic to the image of $X$ under the functor $\operatorname{Tw}( \mathscr {F}(u) ): \operatorname{Tw}( \mathscr {F}(C) ) \rightarrow \operatorname{Tw}( \mathscr {F}(D) )$. The desired result now follows from Example 8.2.1.5, since the functor $\mathscr {F}(u)$ carries isomorphisms in the $\infty $-category $\mathscr {F}(C)$ to isomorphisms in the $\infty $-category $\mathscr {F}(D)$.
$\square$