Kerodon

Remark 8.6.8.15. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an essentially small cocartesian fibration of simplicial sets. Then $U$ is a cartesian fibration if and only if the covariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{QC}}$ factors through the full subcategory $\operatorname{\mathcal{QC}}_{\operatorname{LAdj}} \subset \operatorname{\mathcal{QC}}$. Similarly, if $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ is an essentially small cartesian fibration, then $U'$ is a cocartesian fibration if and only if its contravariant transport representation $\operatorname{Tr}_{\operatorname{\mathcal{E}}'/\operatorname{\mathcal{C}}}: \operatorname{\mathcal{C}}^{\operatorname{op}} \rightarrow \operatorname{\mathcal{QC}}$ factors through the subcategory $\operatorname{\mathcal{QC}}_{\operatorname{RAd}} \subset \operatorname{\mathcal{QC}}$. See Proposition 6.2.3.6.