ScalingStacks

Proof. The first equivalence follows from the definition of $\Psi_{\tri}{\mathcal{A}}$ as the smallest stable subcategory of the stable $\infty$-category $\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ containing the representables, together with the observations that $\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]$ is compactly generated with compact objects

$$\Psi_{\perf}{\mathcal{A}}\simeq\mathrm{N}(\mathrm{Fun}_{\mathcal{S}}({\mathcal{A}}^{\op},{\mathcal{S}})^{\mathrm{c}})[W^{-1}]^{\omega}$$

and $\Ind(\Psi_{\tri}{\mathcal{A}})\simeq\Ind(\Psi_{\perf}{\mathcal{A}})$ (as $\Ind$-categories are automatically idempotent-complete). The second equivalence follows immediately from proposition 3.2. ∎