ScalingStacks

Proof. It is clear from the definition that $\widetilde{U\times E}\approx\widetilde{U}\times\widetilde{E}$, and $\widetilde{E}$ is contractible by (14.2). Thus categorically homotopic maps are taken to homotopic maps by the completion operator, and hence completion takes categorical equivalences to homotopy equivalences. ∎