Kerodon

Remark 7.4.4.13. Let $\lambda $ be an uncountable regular cardinal and let $\kappa = \mathrm{ecf}(\lambda )$ denote its exponential cofinality. In the situation of Proposition 7.4.4.12, suppose that the cocartesian fibration $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ is essentially $\lambda $-small and that the simplicial set $\operatorname{\mathcal{C}}$ is $\kappa $-small. Then fiber of $\overline{U}$ over the cone point of $\operatorname{\mathcal{C}}^{\triangleleft }$ is equivalent to the $\infty $-category $\operatorname{Fun}_{ / \operatorname{\mathcal{C}}}^{\operatorname{CCart}}( \operatorname{\mathcal{C}}, \operatorname{\mathcal{E}})$, which is also essentially $\lambda $-small (Variant 4.7.9.11). It follows that the cocartesian fibration $\overline{U}$ is essentially $\lambda $-small.