Example 4.7.9.5. Let $\kappa $ be an uncountable cardinal and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration of $\infty $-categories. If $\operatorname{\mathcal{E}}$ and $\operatorname{\mathcal{C}}$ are essentially $\kappa $-small, then $U$ is essentially $\kappa $-small. To prove this, we observe that for every $n$-simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$, Proposition 4.6.2.8 supplies a fully faithful functor
It follows from Corollary 4.7.5.16 that the homotopy fiber product $\Delta ^ n \times _{ \operatorname{\mathcal{C}}}^{\mathrm{h}} \operatorname{\mathcal{E}}$ is essentially $\kappa $-small, so that $\operatorname{\mathcal{E}}_{\sigma }$ is also essentially $\kappa $-small (Corollary 4.7.5.14). For a partial converse, see Corollary 4.7.9.12 below.