ScalingStacks

Proof. It suffices to show the result for ${\mathcal{F}}_{\kappa}$, as the statement for $\Sigma_{\kappa}$ follows because colimits commute. Thus, we need to verify that

$$(\Ind_{\omega}({\mathcal{A}}))^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind_{\omega}({\mathcal{B}}))^{\kappa}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}(\Ind_{\omega}({\mathcal{C}}))^{\kappa}$$

is exact. The sequence

$$\Ind_{\omega}{\mathcal{A}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\omega}{\mathcal{B}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\Ind_{\omega}{\mathcal{C}}$$

is exact by Definition 5.12 and Proposition 5.15. Now the result follows from Proposition 5.17. ∎