Kerodon

Proof. Assume first that $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Applying Proposition 4.7.9.5, we deduce that $U$ is locally $\kappa $-small. It will therefore suffice to show that, for each object $C \in \operatorname{\mathcal{C}}$, the $\infty $-category $\operatorname{\mathcal{E}}_{C} = \{ C\} \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small (Remark 4.7.9.4). Using Corollary 4.5.2.23, we can factor $U$ as a composition $\operatorname{\mathcal{E}}\xrightarrow {\iota } \operatorname{\mathcal{E}}' \xrightarrow {U'} \operatorname{\mathcal{C}}$, where $U'$ is an isofibration and $\iota $ is an equivalence of $\infty $-categories. Then $\operatorname{\mathcal{E}}'$ is essentially $\kappa $-small, so Corollary 4.7.5.16 guarantees that the fiber $\operatorname{\mathcal{E}}'_{C}$ is essentially $\kappa $-small. Remark 4.5.2.24 guarantees that the map of fibers $\operatorname{\mathcal{E}}_{C} \rightarrow \operatorname{\mathcal{E}}'_{C}$ is fully faithful, so $\operatorname{\mathcal{E}}_{C}$ is also essentially $\kappa $-small (Corollary 4.7.5.13).

Now suppose that $U$ is an isofibration which is essentially $\kappa $-small; we wish to show that the $\infty $-category $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small. Proposition 4.7.9.5 guarantees that $U$ is locally $\kappa $-small. It will therefore suffice to show that the set of isomorphism classes $\pi _0( \operatorname{\mathcal{E}}^{\simeq } )$ is $\kappa $-small. In fact, we will show that the core $\operatorname{\mathcal{E}}^{\simeq }$ is an essentially $\kappa $-small Kan complex. This is a special case of Corollary 4.7.7.2, since $U$ induces a Kan fibration $U^{\simeq }: \operatorname{\mathcal{E}}^{\simeq } \rightarrow \operatorname{\mathcal{C}}^{\simeq }$ whose fibers are essentially $\kappa $-small (see Proposition 4.4.3.7). $\square$