Remark 8.6.8.18 (05WW)

Remark 8.6.8.18. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cocartesian fibration of simplicial sets, and let $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ denote the covariant transport representation for $U$. Assume that $U$ is also a cartesian fibration, so that $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}$ factors through the subcategory $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}} \subset \operatorname{\mathcal{QC}}$. Then the contravariant transport representation of $U$ is given by the composition

\[ \operatorname{\mathcal{C}}^{\operatorname{op}} \xrightarrow { \operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}^{\operatorname{op}} } \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}^{\operatorname{op}} \xrightarrow { \Psi } \operatorname{\mathcal{QC}}_{\operatorname{RAd}} \subset \operatorname{\mathcal{QC}}, \]

where $\Psi $ is the formation-of-adjoints functor (Construction 8.6.8.16). The proof reduces to the universal case $\operatorname{\mathcal{C}}= \operatorname{\mathcal{QC}}_{\operatorname{LAdj}}$, in which case it is immediate from the definition of $\Psi $.