ScalingStacks

Proof. First, consider the subclass $T^{\prime}\subset T$ containing those morphisms $U^{\prime}\to V^{\prime}$ of $T$ such that for any nondegenerate morphisms $V^{\prime}\to C_{k}$ and $H\to C_{k}$, the pullback

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

lies in $T$. Since colimits in $\pre(\mathcal{R})$ are universal, one deduces immediately that the class $T^{\prime}$ is strongly saturated. Hence (R.1-bis(b)) implies that $T^{\prime}=T$. Thus $T$ is closed under pullbacks along morphisms $H\to C_{k}$ and contains $i^{*}S_{00}$ (by (R.1-bis(a))), hence contains all of $i^{*}S_{0}$. ∎