Corollary 5.2.7.3. Let $\operatorname{\mathcal{C}}$ be a simplicial set, and let $\operatorname{LCov}_{\operatorname{\mathcal{C}}}$ denote the full subcategory of $(\operatorname{Set_{\Delta }})_{/\operatorname{\mathcal{C}}}$ spanned by the left covering maps $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$. Then the formation of homotopy transport representations supplies an equivalence of categories
\[ \operatorname{LCov}_{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) \quad \quad (U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}) \mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}. \]