Proposition 9.4.1.25. Let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category. If the $\infty $-category $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, then the $\infty $-category $\operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{C}})$ is $(\kappa ,\mu )$-compactly generated. The converse holds if $\operatorname{\mathcal{C}}$ is idempotent-complete.
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Proof. If $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated, then it can be identified with $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}_0)$ for some $\infty $-category $\operatorname{\mathcal{C}}_0$ (Proposition 9.4.1.11). Applying Theorem 9.3.6.4, we conclude that $\operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ is equivalent to $\operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}}_0)$, and is therefore $(\kappa ,\mu )$-compactly generated.
We now prove the converse. Assume that $\widehat{\operatorname{\mathcal{C}}} = \operatorname{Ind}_{\lambda }^{\mu }(\operatorname{\mathcal{C}})$ is $(\kappa ,\mu )$-compactly generated: that is, we have an equivalence $\widehat{\operatorname{\mathcal{C}}} \simeq \operatorname{Ind}_{\kappa }^{\mu }(\operatorname{\mathcal{C}}_0)$ for some other $\infty $-category $\operatorname{\mathcal{C}}_0$. If $\operatorname{\mathcal{C}}$ is idempotent-complete, then it can be identified with the full subcategory of $\widehat{\operatorname{\mathcal{C}}}$ spanned by the $(\lambda ,\mu )$-compact objects (Corollary 9.4.1.21). Using Corollary 9.3.6.8, we can identify this $\infty $-category with $\operatorname{Ind}_{\kappa }^{\lambda }(\operatorname{\mathcal{C}}_0)$, so that $\operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compactly generated. $\square$