ScalingStacks

Proof. First note that as $\Theta_{n}$ is dense in $\pre(\Theta_{n})$, and $T_{\Theta_{n}}$ is strongly saturated, it is enough to consider $H\in\Theta_{n}$ representable. Let $f\colon U\to V$ be a morphism in $T_{\Theta_{n}}$, let $V\to C_{i}$ be given, and let $H\to C_{i}$ be arbitrary. There exists a unique factorization $H\to C_{k}\hookrightarrow C_{i}$, with $H\to C_{k}$ nondegenerate. Consider the following diagram of pullbacks in $\pre(\Theta_{n})$:

Since $T^{(c)}_{\Theta_{n}}=T_{\Theta_{n}}$, we have $[U^{\prime}\to V^{\prime}]\in T_{\Theta_{n}}$, and if $T^{(b)}_{\Theta_{n}}=T_{\Theta_{n}}$, then we also have $[U^{\prime\prime}\to V^{\prime\prime}]\in T_{\Theta_{n}}$, as desired. ∎