ScalingStacks

Proof. Recall that ${\mathcal{F}}_{\kappa}$ is the composite (9.3). Hence, the claim follows from the fact that the passage to $\Ind_{\kappa}$ and to $\kappa$-compact objects preserves $\kappa$-filtered colimits [52, 5.5.7.8, 5.5.7.10, 5.5.7.11]. Since $\Sigma_{\kappa}$ is the cofiber of the inclusion ${\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{F}}_{\kappa}$ and colimits commute, we deduce that $\Sigma_{\kappa}$ preserves $\kappa$-filtered colimits if ${\mathcal{F}}_{\kappa}$ does. ∎