How does the geometric Frobenius act on stalk?

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How does the geometric Frobenius act on stalk? The first image is from http://virtualmath1.stanford.edu/~conrad/Weil2seminar/Notes/L19.pdf page 9. The second image is from Kiehl-Weissauer page 7.

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See https://stacks.math.columbia.edu/tag/03QW .

Take special care for as the embedding to closure changes, the gluing in the colimit reflects it.