Corollary 4.7.9.12. Let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an isofibration of $\infty $-categories and let $\kappa $ be an uncountable regular cardinal. If $\operatorname{\mathcal{C}}$ is essentially $\kappa $-small and $U$ is essentially $\kappa $-small, then $\operatorname{\mathcal{E}}$ is essentially $\kappa $-small.
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Proof. Using Proposition 4.7.6.15, we can choose an equivalence of $\infty $-categories $\operatorname{\mathcal{C}}' \rightarrow \operatorname{\mathcal{C}}$, where $\operatorname{\mathcal{C}}'$ is minimal. Since $U$ is an isofibration, Corollary 4.5.2.29 guarantees that the $\infty $-category $\operatorname{\mathcal{E}}' = \operatorname{\mathcal{C}}' \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is equivalent to $\operatorname{\mathcal{E}}$. We may therefore replace $U$ by the projection map $U': \operatorname{\mathcal{E}}' \rightarrow \operatorname{\mathcal{C}}'$, and thereby reduce to the situation where $\operatorname{\mathcal{C}}= \operatorname{\mathcal{C}}'$ is minimal. In this case, $\operatorname{\mathcal{C}}$ is $\kappa $-small (Corollary 4.7.6.12), so the desired result follows from Proposition 4.7.9.10. $\square$