ScalingStacks

Proof. As colimits are universal in $\pre(\Theta_{n})$, the functors $(-)\times_{C_{i}}\nu H$ preserves all small colimits. Thus the class $[(-)\times_{C_{i}}\nu H]^{-1}(T_{\Theta_{n}})$ is a saturated class in $\pre(\Theta_{n})$. Taking appropriate intersections of these classes and $T_{\Theta_{n}}$ yields the three classes in question. ∎