Kerodon

Remark 5.3.1.6. In the situation of Construction 5.3.1.5, suppose that $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{N}_{\bullet }(\operatorname{\mathcal{C}})$ is a left fibration of $\infty $-categories. It follows that every morphism of $\operatorname{\mathcal{E}}$ is $U$-cocartesian (Proposition 5.1.4.15), so the strict transport representation $\operatorname{sTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{QCat}$ coincides with the weak transport representation $\operatorname{wTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$.