Proposition 4.7.9.10. Let $\lambda $ be an uncountable cardinal, let $\kappa = \mathrm{cf}(\lambda )$ be its cofinality, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of simplicial sets. If the simplicial set $\operatorname{\mathcal{C}}$ is $\kappa $-small and $U$ is essentially $\lambda $-small, then the simplicial set $\operatorname{\mathcal{E}}$ is essentially $\lambda $-small.
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Proof. By virtue of Proposition 5.3.8.11, we may assume that the inner fibration $U$ is minimal. Then, for every $n$-simplex $\Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, the fiber product $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is a minimal $\infty $-category which is essentially $\lambda $-small, and is therefore $\lambda $-small (Corollary 4.7.6.12). Applying Remark 4.7.4.9, we conclude that $\operatorname{\mathcal{E}}$ is $\lambda $-small (and therefore essentially $\lambda $-small). $\square$