Kerodon

Warning 4.7.9.13. In the statement of Corollary 4.7.9.12, the assumption that $U$ is an isofibration cannot be omitted. For example, suppose that $\operatorname{\mathcal{C}}= \operatorname{N}_{\bullet }( \operatorname{\mathcal{C}}_0 )$ is the nerve of an essentially $\kappa $-small category $\operatorname{\mathcal{C}}_0$, and let $\operatorname{\mathcal{E}}= \operatorname{sk}_0( \operatorname{\mathcal{C}})$ be the constant simplicial set associated to the collection of objects of $\operatorname{\mathcal{C}}_0$. Then the inclusion map $\operatorname{\mathcal{E}}\hookrightarrow \operatorname{\mathcal{C}}$ is an essentially $\kappa $-small inner fibration (which is usually not an isofibration). However, the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small if and only if the set of objects of $\operatorname{\mathcal{C}}_0$ is $\kappa $-small.