Kerodon

Example 5.3.1.4. Let $\operatorname{\mathcal{C}}$ be a category and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ be a left covering map of simplicial sets. Then, for every object $C \in \operatorname{\mathcal{C}}$, the evaluation map $\operatorname{ev}_{C}: \operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C) \rightarrow \operatorname{\mathcal{E}}_{C}$ is an isomorphism of simplicial sets (Exercise 4.2.5.5). It follows that the simplicial set $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}(C)$ is discrete (see Remark 4.2.3.17). We can therefore identify $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ with a functor from $\operatorname{\mathcal{C}}$ to the category of sets, which is isomorphic to the homotopy transport representation $\operatorname{hTr}_{ \operatorname{\mathcal{E}}/ \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}}) }: \operatorname{\mathcal{C}}\rightarrow \operatorname{Set}$ of Definition 5.2.0.4.