Kerodon

Example 9.3.6.13. Let $\kappa \trianglelefteq \lambda \trianglelefteq \mu $ be regular cardinals and let $\operatorname{\mathcal{C}}$ be an $\infty $-category which is $(\kappa ,\lambda )$-cocomplete. Then an object $C \in \operatorname{\mathcal{C}}$ is $(\kappa ,\lambda )$-compact if and only if its image in $\operatorname{Ind}_{\lambda }^{\mu }( \operatorname{\mathcal{C}})$ is $(\kappa ,\mu )$-compact. This follows by applying Corollary 9.3.6.11 to the functor $h^{C}: \operatorname{\mathcal{C}}\rightarrow \operatorname{\mathcal{S}}_{< \nu }$ corepresented by $C$ (for some sufficiently large cardinal $\nu $).