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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