Proposition 4.7.9.15. Let $\kappa $ be an uncountable regular cardinal, let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is locally $\kappa $-small, and let $U: \operatorname{\mathcal{E}}\rightarrow \operatorname{\mathcal{C}}$ be an inner fibration. The following conditions are equivalent:
The inner fibration $U$ is locally $\kappa $-small.
The $\infty $-category $\operatorname{\mathcal{E}}$ is locally $\kappa $-small.
Proof. Assume first that $(1)$ is satisfied; we will prove $(2)$. Let $X$ and $Y$ be objects of $\operatorname{\mathcal{E}}$, and set $\overline{X} = U(X)$ and $\overline{Y} = U(Y)$. We wish to show that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. By virtue of Proposition 4.6.1.21, the functor $U$ induces a Kan fibration $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$. Our assumption that $\operatorname{\mathcal{C}}$ is locally $\kappa $-small guarantees that the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$ is essentially $\kappa $-small. By virtue of Corollary 4.7.7.2, it will suffice to show that for every morphism $e: \overline{X} \rightarrow \overline{Y}$ in $\operatorname{\mathcal{C}}$, the Kan complex $\operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)_{e} = \{ e \} \times _{ \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} ) } \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y)$ is essentially $\kappa $-small. This follows immediately from the local $\kappa $-smallness of the $\infty $-category $\operatorname{\mathcal{E}}_{e} = \Delta ^1 \times _{\operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$.
We now show that $(2)$ implies $(1)$. Assume that $\operatorname{\mathcal{E}}$ is locally $\kappa $-small and choose a simplex $\sigma : \Delta ^ n \rightarrow \operatorname{\mathcal{C}}$; we will show that the $\infty $-category $\operatorname{\mathcal{E}}_{\sigma } = \Delta ^ n \times _{ \operatorname{\mathcal{C}}} \operatorname{\mathcal{E}}$ is locally $\kappa $-small. Fix a pair of objects $\widetilde{X}, \widetilde{Y} \in \operatorname{\mathcal{E}}_{\sigma }$; we wish to show that the Kan complex $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{\sigma } }( \widetilde{X}, \widetilde{Y} )$ is essentially $\kappa $-small. Let $X$ and $Y$ denote the images of $\widetilde{X}$ and $\widetilde{Y}$ in the $\infty $-category $\operatorname{\mathcal{E}}$, and set $\overline{X} = U(X)$ and $\overline{Y} = U(Y)$ as above. If the Kan complex $\operatorname{Hom}_{ \operatorname{\mathcal{E}}_{\sigma } }( \widetilde{X}, \widetilde{Y} )$ is nonempty, then it can be identified with a fiber of the Kan fibration $\theta : \operatorname{Hom}_{\operatorname{\mathcal{E}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}( \overline{X}, \overline{Y} )$, which is essentially $\kappa $-small by virtue of Corollary 4.7.7.2. $\square$