Lemma 65.16.5. The base change of an étale morphism of algebraic spaces by any morphism of algebraic spaces is étale.

**Proof.**
Let $X \to Y$ be an étale morphism of algebraic spaces over $S$. Let $Z \to Y$ be a morphism of algebraic spaces. Choose a scheme $U$ and a surjective étale morphism $U \to X$. Choose a scheme $W$ and a surjective étale morphism $W \to Z$. Then $U \to Y$ is étale, hence in the diagram

the top horizontal arrow is étale. Moreover, the left vertical arrow is surjective and étale (verification omitted). Hence we conclude that the lower horizontal arrow is étale by Lemma 65.16.3. $\square$

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