ScalingStacks

Proof. This follows from the following equivalences

(9.18)		$\displaystyle I\mspace{-6.mu}K({\mathcal{A}})$	$\displaystyle=$	$\displaystyle\underset{n\geq 0}{\mathrm{hocolim}}\,\Omega^{n}K(\Sigma_{\kappa}^{(n)}({\mathcal{A}}))$
			$\displaystyle\simeq$	$\displaystyle\underset{n\geq 0}{\mathrm{hocolim}}\,\Omega^{n}\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\,\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}})))$
(9.19)			$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty})^{\kappa}),\,\underset{n\geq 0}{\mathrm{colim}}\,\Sigma^{-n}\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}(\Sigma_{\kappa}^{(n)}({\mathcal{A}})))$
			$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\kappa}),\,V({\mathcal{A}}))\,.$

Equivalence (9.18) comes from theorem 9.10 and equivalence (9.19) comes from the compactness of $\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ in $\underline{{\mathcal{M}}_{\mathrm{wloc}}^{\kappa}}$. ∎