Remark 7.5.3.9. Let $\operatorname{\mathcal{C}}$ be a category, let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a morphism of simplicial sets, and let $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set_{\Delta }}$ denote the weak transport representation of Construction 5.3.1.1, given on objects by the formula $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) = \operatorname{Fun}_{ / \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_{C/} ), \operatorname{\mathcal{E}})$. Then Lemma 7.5.3.8 supplies a canonical isomorphism of simplicial sets
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
\[ \operatorname{Fun}_{ / \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }( \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}), \operatorname{\mathcal{E}}) \xrightarrow {\sim } \varprojlim ( \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}} ). \]