The Stacks project

Lemma 6.21.10. Suppose that $f : X \to Y$ and $g : Y \to Z$ are continuous maps of topological spaces. Suppose that $\mathcal{F}$ is a sheaf on $X$, $\mathcal{G}$ is a sheaf on $Y$, and $\mathcal{H}$ is a sheaf on $Z$. Let $\varphi : \mathcal{G} \to \mathcal{F}$ be an $f$-map. Let $\psi : \mathcal{H} \to \mathcal{G}$ be an $g$-map. Let $x \in X$ be a point. The map on stalks $(\varphi \circ \psi )_ x : \mathcal{H}_{g(f(x))} \to \mathcal{F}_ x$ is the composition

\[ \mathcal{H}_{g(f(x))} \xrightarrow {\psi _{f(x)}} \mathcal{G}_{f(x)} \xrightarrow {\varphi _ x} \mathcal{F}_ x \]

Proof. Immediate from Definition 6.21.9 and the definition of the map on stalks above. $\square$

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