Lemma 6.22.1 (008O)—The Stacks project

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Table of contents
Part 1: Preliminaries
Chapter 6: Sheaves on Spaces
Section 6.22: Continuous maps and abelian sheaves
Lemma 6.22.1 (cite)

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Lemma 6.22.1. Let $f : X \to Y$ be a continuous map.

Let $\mathcal{G}$ be an abelian presheaf on $Y$ . Let $x \in X$ . The bijection $\mathcal{G}_{f(x)} \to (f_ p\mathcal{G})_ x$ of Lemma 6.21.4 is an isomorphism of abelian groups.
Let $\mathcal{G}$ be an abelian sheaf on $Y$ . Let $x \in X$ . The bijection $\mathcal{G}_{f(x)} \to (f^{-1}\mathcal{G})_ x$ of Lemma 6.21.5 is an isomorphism of abelian groups.

Proof. Omitted. $\square$

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