ScalingStacks

Theorem 7.8. Let ${\mathcal{A}}$ be a simplicial model category and ${\mathcal{C}}\subset{\mathcal{A}}$ a small full subcategory of the cofibrants which admits all homotopy pushouts and is a Waldhausen category via the model structure on ${\mathcal{A}}$. Then there is an equivalence of spectra

$$K({\mathcal{C}})\simeq K(\mathrm{N}(({\mathcal{C}})^{\cf}))$$

which is natural in weakly exact functors.