Kerodon

Definition 8.1.9.5. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets and let $\overline{f}: \overline{X} \rightarrow \overline{Y}$ be an edge of $\operatorname{\mathcal{C}}$. We say that $\overline{f}$ admits $U$-cartesian lifts if, for every vertex $Y \in \operatorname{\mathcal{E}}$ satisfying $U(Y) = \overline{Y}$, there is a $U$-cartesian edge $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$. We say that $\overline{f}$ admits $U$-cocartesian lifts if, for every vertex $X \in \operatorname{\mathcal{E}}$ satisfying $U(X) = \overline{X}$, there is a $U$-cocartesian edge $f: X \rightarrow Y$ of $\operatorname{\mathcal{E}}$ satisfying $U(f) = \overline{f}$.