ScalingStacks

Proof. Let us focus on the case $x=a$. Let $f\colon U\to V$ be in class given on the right-hand side of the asserted identity. We wish to show that $f\in T_{\Theta_{n}}^{(a)}$, that is for any pair of morphism $H\to C_{i}$ and $V\to C_{i}$ we have

$$U^{\prime}=U\times_{C_{i}}H\to V\times_{C_{i}}H=V^{\prime}$$

is in $T_{\Theta_{n}}$. This follows as there exists a factorization $V\to C_{k}\to C_{i}$ and a diagram of pullbacks:

such that $V\to C_{k}$ is nondegenerate. The analogous result for $T_{\Theta_{n}}^{(b)}$ follows by the same argument and the observation that $H^{\prime}\to C_{k}$ is nondegenerate if $H\to C_{i}$ is such. ∎