Kerodon

Remark 5.3.1.13. We will be primarily interested in the special case of Notation 5.3.1.10 where $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}$ is a left fibration of simplicial sets. In this case, an object $F \in \operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ belongs to the full subcategory $\operatorname{Fun}^{\operatorname{CCart}}_{/\operatorname{\mathcal{C}}}( \operatorname{\mathcal{E}}', \operatorname{\mathcal{E}})$ if and only if it carries every edge of $\operatorname{\mathcal{E}}'$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$ (Proposition 5.1.4.14).