Kerodon

Warning 5.2.5.8. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a morphism of simplicial sets which is both a cartesian fibration and a cocartesian fibration. Then Constructions 5.2.5.2 and 5.2.5.7 supply functors $\mathrm{h} \mathit{\operatorname{\mathcal{C}}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ and $\mathrm{h} \mathit{\operatorname{\mathcal{C}}}^{\operatorname{op}} \rightarrow \mathrm{h} \mathit{\operatorname{QCat}}$ respectively, which are both referred to as the homotopy transport representation of $U$ and both denoted by $\operatorname{hTr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$. We will see later that these two functors are interchangeable data: either can be recovered from the other (see Proposition 6.2.3.6).