ScalingStacks

Proof. If a categorical homotopy between $f$ and $g$ is given by $H\colon A\times E\rightarrow B$, then $W^{H}\colon W^{B}\rightarrow W^{A\times E}\approx(W^{A})^{E}$ is a categorical homotopy of $W^{f}$ and $W^{g}$. The statement about categorical equivalences follows. ∎