The Stacks project

Lemma 63.6.2. Let $j : U \to X$ be an étale morphism. Then $j^! = j^{-1}$.

Proof. This is true because $j_!$ as defined in Section 63.4 agrees with $j_!$ as defined in Étale Cohomology, Section 59.70, see Lemma 63.4.3. Finally, in Étale Cohomology, Section 59.70 the functor $j_!$ is defined as the left adjoint of $j^{-1}$ and hence we conclude by uniqueness of adjoint functors. $\square$

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