0NPZ Proof. Let π{\mathcal{D}} be an idempotent-complete stable β\infty-category. Then Funexβ(πΟβ[Ξ£β1],π)\displaystyle\mathrm{Fun}^{\ex}(\mathcal{C}^{\omega}[\Sigma^{-1}],{\mathcal{D}}) βlimFunexβ(πΟ,π)βlimFunΟLβ(π,Indβ‘(π))\displaystyle\simeq\lim\mathrm{Fun}^{\ex}(\mathcal{C}^{\omega},{\mathcal{D}})\simeq\lim\mathrm{Fun}^{\mathrm{L}}_{\omega}(\mathcal{C},\Ind({\mathcal{D}})) βFunΟLβ(Stabβ‘(π),Indβ‘(π))βFunexβ(Stabβ‘(π)Ο,π).\displaystyle\simeq\mathrm{Fun}^{\mathrm{L}}_{\omega}(\Stab(\mathcal{C}),\Ind({\mathcal{D}}))\simeq\mathrm{Fun}^{\mathrm{\ex}}(\Stab(\mathcal{C})^{\omega},{\mathcal{D}}). Since Stabβ‘(π)Ο\Stab(\mathcal{C})^{\omega} is necessarily idempotent-complete, we conclude that it is equivalent to the idempotent-completion of πΟβ[Ξ£β1]\mathcal{C}^{\omega}[\Sigma^{-1}]. β