Lemma 114.8.23. Let $f : X \to Y$ be a locally quasi-finite morphism of schemes. There exists a unique functor $f^! : \textit{Ab}(Y_{\acute{e}tale}) \to \textit{Ab}(X_{\acute{e}tale})$ such that

1. for any open $j : U \to X$ with $f \circ j$ separated there is a canonical isomorphism $j^! \circ f^! = (f \circ j)^!$, and

2. these isomorphisms for $U \subset U' \subset X$ are compatible with the isomorphisms in More Étale Cohomology, Lemma 62.6.3.

Proof. Immediate consequence of More Étale Cohomology, Lemmas 62.6.1 and 62.6.3. $\square$

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