Proposition 8.6.4.21. Let $\operatorname{\mathcal{C}}$ be a category and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \mathbf{Cat}$ be a functor of $2$-categories. Then the functor
\[ \operatorname{N}_{\bullet }( \mathscr {K} ): \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F}') \times _{ \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) } \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{Set}) \subset \operatorname{\mathcal{S}} \]
of Construction 8.6.4.20 exhibits the projection map $U': \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F}' ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$ as a cocartesian dual of the projection map $U: \operatorname{N}_{\bullet }( \int _{\operatorname{\mathcal{C}}} \mathscr {F} ) \rightarrow \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}})$.
Proof.
For each object $C \in \operatorname{\mathcal{C}}$, the restriction of $\mathscr {K}$ to the fiber over $C$ is given concretely by the functor
\[ \mathscr {K}_{C}: \mathscr {F}(C)^{\operatorname{op}} \times \mathscr {F}(C) \rightarrow \operatorname{Set}\quad \quad (X',X) \mapsto \operatorname{Hom}_{ \mathscr {F}(C) }(X',X). \]
Example 8.3.3.4 implies that $\operatorname{N}_{\bullet }( \mathscr {K}_{C} )$ is a $\operatorname{Hom}$-functor for the $\infty $-category $\operatorname{N}_{\bullet }( \mathscr {F}(C) )$ and is therefore a balanced profunctor (Proposition 8.3.3.8). Let $u: C \rightarrow D$ be a morphism in the category $\operatorname{\mathcal{C}}$, and let $f: (C,X) \rightarrow (D,Y)$ and $f': (C,X') \rightarrow (D,Y')$ be lifts of $u$ to the categories $\int _{\operatorname{\mathcal{C}}} \mathscr {F}$ and $\int _{\operatorname{\mathcal{C}}} \mathscr {F}'$, respectively. We wish to show that, if $f$ is $U$-cocartesian and $f'$ is $U'$-cocartesian, then the induced map
\[ \mathscr {K}(f,f'): \mathscr {K}_{C}( X',X ) \rightarrow \mathscr {K}_{D}( Y', Y ) \]
carries universal elements of $\mathscr {K}_{C}(X',X)$ to universal elements of $\mathscr {K}_{D}(Y',Y)$. Let us identify $f$ and $f'$ with morphisms $g: \mathscr {F}(u)(X) \rightarrow Y$ and $g': Y' \rightarrow \mathscr {F}(u)(Y)$ in the category $\mathscr {F}(D)$, so that $\mathscr {K}(f',f)$ is given by the composition
\[ \operatorname{Hom}_{ \mathscr {F}(C) }( X', X) \xrightarrow { \mathscr {F}(u) } \operatorname{Hom}_{ \mathscr {F}(D) }( \mathscr {F}(u)(X'), \mathscr {F}(u)(X) ) \xrightarrow { g \circ - \circ g' } \operatorname{Hom}_{ \mathscr {F}(D) }( Y', Y ). \]
Our assumption that $f$ is $U$-cocartesian guarantees that $g$ is an isomorphism in the category $\mathscr {F}(D)$, and our assumption that $f'$ is a $U'$-cocartesian guarantees that $g'$ is an isomorphism in the category $\mathscr {F}(D)$. The desired result now follows from the observation that if $e: X' \rightarrow X$ is an isomorphism in the category $\mathscr {F}(C)$, then the composition $g \circ \mathscr {F}(u)(e) \circ g'$ is an isomorphism in the category $\mathscr {F}(D)$.
$\square$