Proposition 9.4.6.5. Let $\kappa \trianglelefteq \lambda $ be small regular cardinals. Then every $\kappa $-accessible $\infty $-category $\operatorname{\mathcal{C}}$ is also $\lambda $-accessible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By assumption, $\operatorname{\mathcal{C}}$ can be identified with the $\operatorname{Ind}_{\kappa }$-completion of an essentially small $\infty $-category $\operatorname{\mathcal{C}}_0$. Applying Theorem 9.3.6.4, we conclude that $\operatorname{\mathcal{C}}$ is also an $\operatorname{Ind}_{\lambda }$-completion of the $\infty $-category $\operatorname{\mathcal{C}}_1 = \operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}_0)$, which is essentially small by virtue of Variant 9.3.3.3. $\square$