ScalingStacks

Proof. Write ${\mathcal{A}}\simeq\colim_{\alpha}{\mathcal{A}}_{\alpha}$ as a $\kappa$-filtered colimit of $\kappa$-compact small stable $\infty$-categories ${\mathcal{A}}_{\alpha}$. Then $\Idem({\mathcal{A}})\simeq\colim_{\alpha}\Idem({\mathcal{A}}_{\alpha})$, since $\Idem$ (viewed as an endofunctor of $\Cat_{\infty}^{\ex}$) commutes with $\kappa$-filtered colimits — this follows from the characterization of $\Idem$ in terms of a subcategory of the $\Ind$ category [52, 5.4.2.4] and the fact that filtered colimits in $\Cat_{\infty}^{\ex}$ can be computed in $\Cat_{\infty}$ [53, 1.1.4.6]. ∎