ScalingStacks

Proof. The additivity theorem implies that, for Waldhausen categories with factorization, the suspension endomorphism $\Sigma:{\mathcal{C}}\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}{\mathcal{C}}$ induces $-\id:K({\mathcal{C}})\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}K({\mathcal{C}})$. By naturality, we conclude that $\Sigma:\mathrm{N}({\mathcal{C}})[W^{-1}])\mathchoice{\longrightarrow}{\rightarrow}{\rightarrow}{\rightarrow}\mathrm{N}({\mathcal{C}})[W^{-1}]$ acts invertibly on $K$-theory. Finally, since $K$-theory (viewed as a functor of small $\infty$-categories with finite colimits and a zero object and right-exact functors) preserves filtered colimits, we see that

$$K(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])\simeq\colim K(\mathrm{N}({\mathcal{C}})[W^{-1}])\simeq K(\mathrm{N}({\mathcal{C}})[W^{-1}])\simeq K({\mathcal{C}}),$$

where the last equivalence follows from Corollary 7.12. ∎