Kerodon

Remark 5.3.1.20. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets. Let $\operatorname{\mathcal{E}}^{\circ } \subseteq \operatorname{\mathcal{E}}$ be the simplicial subset whose $n$-simplices are maps $\Delta ^{n} \rightarrow \operatorname{\mathcal{E}}$ which carry each edge of $\Delta ^ n$ to a $U$-cocartesian edge of $\operatorname{\mathcal{E}}$, so that $U$ restricts to a left fibration $U^{\circ }: \operatorname{\mathcal{E}}^{\circ } \rightarrow \operatorname{\mathcal{C}}$ (see Corollary 5.1.4.15). Then $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}}^{\circ } )$ can be identified with the core of the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$.