Kerodon

Remark 5.2.5.4. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets, and let $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ be the homotopy transport representation of Construction 5.2.5.2. It follows from Proposition 5.1.4.15 that $U$ is a left fibration if and only if the functor $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ factors through the full subcategory $\mathrm{h} \mathit{\operatorname{Kan}} \subseteq \mathrm{h} \mathit{\operatorname{QCat}}$. In particular, if $U$ is a left fibration, then Construction 5.2.5.2 determines a functor $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{Kan}}$ which we will also refer to as the homotopy transport representation of the left fibration $U$.