Kerodon

Example 5.2.5.3. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering map of simplicial sets. Then the homotopy transport representation $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ of Construction 5.2.5.2 coincides with the functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ of Proposition 5.2.0.3 (here we abuse notation by identifying the category of sets with the full subcategory of $\mathrm{h} \mathit{\operatorname{Kan}}$ spanned by the discrete simplicial sets).