Corollary 5.2.7.6. Let $\operatorname{\mathcal{C}}$ be a Kan complex. Then the construction $(U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}) \mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ induces an equivalence of categories
\[ \{ \textnormal{Covering maps $\operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$} \} \rightarrow \operatorname{Fun}( \pi _{\leq 1}(\operatorname{\mathcal{C}}), \operatorname{Set}). \]