Warning 8.6.8.6. We have now attached two meanings to the notation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$:
If $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is an (essentially small) cartesian fibration, then $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ can denote the contravariant transport representation of $U$, regarded as an object of $\operatorname{Fun}( \operatorname{\mathcal{C}}^{\operatorname{op}}, \operatorname{\mathcal{QC}})$ (Notation 8.6.8.5).
If $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is an (essentially small) cocartesian fibration, then $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ can denote the covariant transport representation of $U$, regarded as an object of $\operatorname{Fun}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{QC}})$ (Notation 5.6.5.16).
In the case where $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is both a cartesian and cocartesian fibration, these usages are in conflict. However, in this case there is a simple relationship between the covariant and contravariant transport representations of $U$: see Remark 8.6.8.18.