ScalingStacks

Proof. By construction, the object $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ is compact in $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$. Let $S$ denote the set of maps in (8.4), $\overline{S}$ the strongly saturated collection of arrows generated by $S$ [52, 5.5.4.5], and let $X$ be an $S$-local object such that the map $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ is an $S$-local equivalence (i.e., $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{A}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}X$ is in $\overline{S}$). Then by definition,

$$\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),\underline{{\mathcal{U}}_{\mathrm{wloc}}^{\kappa}}({\mathcal{A}}))\simeq\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),X),$$

so it suffices to show that the functor

(9.11)

$$R:=\mathrm{Map}(\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega}),-):\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}\longrightarrow{\mathcal{S}}_{\infty}$$

sends the maps in $\overline{S}$ to equivalences of spectra. Since $\underline{{\mathcal{M}}_{\mathrm{add}}^{\kappa}}$ is a stable $\infty$-category and $\underline{{\mathcal{U}}_{\mathrm{add}}^{\kappa}}({\mathcal{S}}_{\infty}^{\omega})$ is compact, $R$ preserves small colimits, so the two-out-of-three property allows us to reduce to checking that $R$ sends the elements of $S$ to equivalences.