# Kerodon

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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

$\operatorname{LCov}(\operatorname{\mathcal{C}}) \rightarrow \operatorname{Fun}( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) \quad \quad (U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}) \mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}.$

Proof. Proposition 5.2.7.2 shows that the functor

$( \mathscr {F} \in \operatorname{Fun}(\mathrm{h} \mathit{\operatorname{\mathcal{C}}}, \operatorname{Set}) ) \mapsto \operatorname{\mathcal{C}}\times _{\operatorname{N}_{\bullet }( \mathrm{h} \mathit{\operatorname{\mathcal{C}}}) } \operatorname{N}_{\bullet }(\int _{\mathrm{h} \mathit{\operatorname{\mathcal{C}}}} \mathscr {F}) \in \operatorname{LCov}(\operatorname{\mathcal{C}})$

is a left homotopy inverse to the functor $\operatorname{\mathcal{E}}\mapsto \operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. By virtue of Example 5.2.0.6 and Remark 5.2.5.6, it is also a right homotopy inverse. $\square$