$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Corollary 5.3.2.23. Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a morphism of simplicial sets, and let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor. Then composition with the natural transformation $u_{\mathscr {F}}$ of Construction 5.3.2.20 induces an isomorphism of simplicial sets
\[ \operatorname{Fun}_{/ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}}) \rightarrow \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} )_{\bullet }, \]
where $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ is the weak transport representation of Construction 5.3.1.1.
Proof.
Define $\mathscr {G}, \mathscr {H}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ by the formulae $\mathscr {G}( C) = \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}})$ and $\mathscr {H}(C) = \operatorname{Fun}( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) )$. We have a commutative diagram of simplicial sets
\[ \xymatrix@R =50pt@C=50pt{ \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{\mathcal{E}}) \ar [r] \ar [d]^{U \circ } & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {G} )_{\bullet } \ar [d]^{U \circ } \\ \operatorname{Fun}( \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ), \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) ) \ar [r] & \operatorname{Hom}_{ \operatorname{Fun}(\operatorname{\mathcal{C}}, \operatorname{Set_{\Delta }}) }( \mathscr {F}, \mathscr {H} )_{\bullet }, } \]
where the horizontal maps are isomorphisms by virtue of Proposition 5.3.2.21. Corollary 5.3.2.23 follows by restricting to fibers of the vertical maps.
$\square$