ScalingStacks

Proof. The result follows from the analogue of the argument for lemma 6.4 in the context of $\kappa$-compact objects and $\Ind_{\kappa}$, and from the universal property of Bousfield localization (see section 2.5 the functor $\psi$ preserves $\kappa$-filtered colimits and proposition 5.27 shows that any split-exact sequence can be approximated by a $\kappa$-filtered colimit of split-exact sequences in $\underline{{\mathcal{E}}_{\mathrm{A}}^{\kappa}}$). ∎