Lemma 76.17.5. Same notation and assumptions as in Lemma 76.17.4. We identify sheaves on $X$ and $X'$ via (76.9.1.1). There is an action of the sheaf

on the sheaf (76.17.4.1). Moreover, the action is simply transitive for any object $U'$ of $(X')_{spaces, {\acute{e}tale}}$ over which the sheaf (76.17.4.1) has a section.

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