Kerodon

Remark 7.4.5.17. Let $\lambda $ be an uncountable cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be a cocartesian fibration of simplicial sets which is essentially $\lambda $-small. Let $\kappa = \mathrm{cf}(\lambda )$ be the cofinality of $\lambda $. If $\operatorname{\mathcal{C}}$ is $\kappa $-small, then the simplicial set $\operatorname{\mathcal{E}}$ is essentially $\lambda $-small (Proposition 4.7.9.10), so any localization of $\operatorname{\mathcal{E}}$ is also essentially $\lambda $-small (Variant 6.3.2.6). It follows that any cocartesian fibration $\overline{U}: \overline{\operatorname{\mathcal{E}}} \rightarrow \operatorname{\mathcal{C}}^{\triangleright }$ satisfying the requirements of Proposition 7.4.5.16 is also essentially $\lambda $-small.