Kerodon

Proof. Assume that, for every vertex $s \in S$, the $\infty $-category $X_{s}$ is locally $\kappa $-small; we will show that $q$ is locally $\kappa $-small (the reverse implication is immediate from the definitions). By virtue of Corollary 4.7.9.11, we may assume without loss of generality that $S = \Delta ^1$. In this case, $q$ is a cartesian fibration and we wish to show that $X$ is locally $\kappa $-small. Fix a pair of objects $x,z \in X$; we wish to show that the Kan complex $\operatorname{Hom}_{ X }(x,z)$ is essentially $\kappa $-small. We may assume without loss of generality that $q(x) \leq q(z)$ (otherwise, the Kan complex $\operatorname{Hom}_{X}(x,z)$ is empty). If $q(x) = q(z)$, then the desired result follows from our hypothesis. It will therefore suffice to treat the case where $q(x) = 0$ and $q(z) = 1$. Since $q$ is a locally cartesian fibration, we can choose a $q$-cartesian morphism $f: y \rightarrow z$ of $X$, where $q(y) = 0$. In this case, composition with the homotopy class $[f]$ induces a homotopy equivalence of Kan complexes $\operatorname{Hom}_{X}(x,y) \rightarrow \operatorname{Hom}_{X}(x,z)$ (Corollary 5.1.2.3), so the desired result follows from the local $\kappa $-smallness of the $\infty $-category $X_{0}$. $\square$