ScalingStacks

Proof. It suffices to show that the mapping spaces in $\Cat_{\mathcal{T}}^{\ex}$ have the correct homotopy type, and this follows from the comparison between the mapping spaces of $\Cat_{\mathcal{T}}$ and $\Cat_{\infty}$ [52, 2.2.0.1] and the fact that on both sides we define the mapping spaces by the same restriction of vertices. ∎