Definition 5.2.0.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a left covering morphism of simplicial sets and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \operatorname{Set}$ be the functor of Proposition 5.2.0.3. We will refer to $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ as the homotopy transport representation of $U$.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$