Kerodon

Example 4.7.9.4. Let $\kappa $ be an uncountable cardinal. Then an $\infty $-category $\operatorname{\mathcal{E}}$ is locally $\kappa $-small (in the sense of Definition 4.7.8.1) if and only if the inner fibration $U: \operatorname{\mathcal{E}}\rightarrow \Delta ^0$ is locally $\kappa $-small (in the sense of Definition 4.7.9.1). Similarly, $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small (in the sense of Definition 4.7.5.1) if and only if $U$ is essentially $\kappa $-small.