Lemma 74.10.2. Let $S$ be a scheme. Let $\tau \in \{ fpqc, fppf, syntomic, smooth, {\acute{e}tale}\} $. Let $\mathcal{P}$ be a property of morphisms of algebraic spaces over $S$ which is $\tau $ local on the target. Let $f : X \to Y$ have property $\mathcal{P}$. For any morphism $Y' \to Y$ which is flat, resp. flat and locally of finite presentation, resp. syntomic, resp. étale, the base change $f' : Y' \times _ Y X \to Y'$ of $f$ has property $\mathcal{P}$.

**Proof.**
This is true because we can fit $Y' \to Y$ into a family of morphisms which forms a $\tau $-covering.
$\square$

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