ScalingStacks

Proof. The proof is analogous to the argument for theorem 7.13; instead of the idempotent-complete stable $\infty$-category $\mathrm{Fun}^{\ex}({\mathcal{B}},\Idem({\mathcal{A}}))$ we consider the small stable $\infty$-category $\mathrm{Fun}^{\ex}({\mathcal{B}},{\mathcal{A}})$. Note that since $\kappa>\omega$, ${\mathcal{S}}_{\infty}^{\omega}$ belongs to $(\Cat_{\infty}^{\ex})^{\kappa}$. ∎