$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proposition 7.5.3.1. Let $\operatorname{\mathcal{C}}$ be a small category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be an isofibration of $\infty $-categories. Then the weak transport representation
\[ \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}\quad \quad C \mapsto \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}}) \]
of Construction 5.3.1.1 is an isofibrant diagram of simplicial sets.
Proof.
Let $\mathscr {F}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ be a functor and let $\mathscr {F}_0 \subseteq \mathscr {F}$ be a subfunctor for which the inclusion $\mathscr {F}_0 \hookrightarrow \mathscr {F}$ is a levelwise categorical equivalence. We wish to show that every natural transformation $\alpha _0: \mathscr {F}_0 \rightarrow \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ admits an extension $\alpha : \mathscr {F} \rightarrow \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. Using Corollary 5.3.2.23, we can reformulate this as a lifting problem
\[ \xymatrix@R =50pt@C=50pt{ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_0 ) \ar [r] \ar [d] & \operatorname{\mathcal{E}}\ar [d]^{U} \\ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} ) \ar [r] \ar@ {-->}[ur] & \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) } \]
in the category of simplicial sets. Since $U$ is an isofibration, we are reduced to showing that the inclusion map $ \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F}_0 ) \hookrightarrow \underset { \longrightarrow }{\mathrm{holim}}( \mathscr {F} )$ is a categorical equivalence (Proposition 4.5.5.1), which is a special case of Variant 5.3.2.19.
$\square$