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Corollary 5.1.5.15. Let $\kappa $ be an uncountable cardinal and let $q: X \rightarrow S$ be a locally cartesian fibration of simplicial sets. Then $q$ is essentially $\kappa $-small if and only if, for every vertex $s \in S$, the $\infty $-category $X_{s} = \{ s\} \times _{S} X$ is essentially $\kappa $-small.

Proof. Without loss of generality, we may assume that $S = \Delta ^ n$ is a standard simplex. Using Corollary 4.7.6.17, we can reduce to the case where $\kappa $ is regular. In this case, the result follows by combining Proposition 5.1.5.14 with Remark 4.7.9.6. $\square$