Lemma 63.6.4. Let $j : U \to X$ and $j' : V \to U$ be étale morphisms. The isomorphism $(j \circ j')^{-1} = (j')^{-1} \circ j^{-1}$ and the isomorphism $(j \circ j')^! = (j')^! \circ j^!$ of Lemma 63.6.3 agree via the isomorphism of Lemma 63.6.2.

**Proof.**
Omitted.
$\square$

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