ScalingStacks

Proof. This follows from the following equivalences

(9.27)		$\displaystyle\mathrm{Map}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{S}}_{\infty}^{\omega}),{\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}}))$	$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\gamma^{\ast}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})))$
			$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\mathrm{Loc}({\mathcal{U}}_{\mathrm{loc}}^{\kappa}({\mathcal{A}})))$
(9.28)			$\displaystyle\simeq$	$\displaystyle\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),V({\mathcal{A}}))$
(9.29)			$\displaystyle\simeq$	$\displaystyle I\mspace{-6.mu}K({\mathcal{A}})\,.$

Equivalence (9.27) comes from proposition 9.25, equivalence (9.28) comes from corollary 9.24, and equivalence (9.29) is proposition 9.17. ∎