Lemma 99.3.5. Let $P$ be a property of morphisms of algebraic spaces as above. Let $\tau \in \{ {\acute{e}tale}, smooth, syntomic, fppf\}$. Let $\mathcal{X} \to \mathcal{Y}$ and $\mathcal{Y} \to \mathcal{Z}$ be morphisms of algebraic stacks representable by algebraic spaces. Assume

1. $\mathcal{X} \to \mathcal{Y}$ is surjective and étale, smooth, syntomic, or flat and locally of finite presentation,

2. the composition has $P$, and

3. $P$ is local on the source in the $\tau$ topology.

Then $\mathcal{Y} \to \mathcal{Z}$ has property $P$.

Proof. Let $Z$ be a scheme and let $Z \to \mathcal{Z}$ be a morphism. Set $X = \mathcal{X} \times _\mathcal {Z} Z$, $Y = \mathcal{Y} \times _\mathcal {Z} Z$. By (1) $\{ X \to Y\}$ is a $\tau$ covering of algebraic spaces and by (2) $X \to Z$ has property $P$. By (3) this implies that $Y \to Z$ has property $P$ and we win. $\square$

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).