ScalingStacks

Proof. This follows from the natural equivalence $I\mspace{-6.mu}K(-)\simeq I\mspace{-6.mu}K(\mathrm{N}(-)[W^{-1}][\Sigma^{-1}])$ and the fact that cofiber sequence

$$I\mspace{-6.mu}K(\mathrm{N}({\mathcal{A}})[W^{-1}][\Sigma^{-1}])\longrightarrow I\mspace{-6.mu}K(\mathrm{N}({\mathcal{B}})[W^{-1}][\Sigma^{-1}])\longrightarrow I\mspace{-6.mu}K(\mathrm{N}({\mathcal{C}})[W^{-1}][\Sigma^{-1}])$$

is a cofiber sequence because $I\mspace{-6.mu}K(-)$ is a localizing invariant. ∎