Lemma 41.14.3. Let $U \to X$ be an étale morphism of schemes where $X$ is a scheme in characteristic $p$. Then the relative Frobenius $F_{U/X} : U \to U \times _{X, F_ X} X$ is an isomorphism.

**Proof.**
The morphism $F_{U/X}$ is a universal homeomorphism by Varieties, Lemma 33.36.6. The morphism $F_{U/X}$ is étale as a morphism between schemes étale over $X$ (Morphisms, Lemma 29.36.18). Hence $F_{U/X}$ is an isomorphism by Theorem 41.14.1.
$\square$

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