Lemma 18.43.3. Let $\mathcal{C}$ be a site with a final object $X$.

1. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of sets on $\mathcal{C}$. If $\mathcal{F}$ is finite locally constant, there exists a covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant sheaves associated to a map of sets.

2. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of abelian groups on $\mathcal{C}$. If $\mathcal{F}$ is finite locally constant, there exists a covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant abelian sheaves associated to a map of abelian groups.

3. Let $\Lambda$ be a ring. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of locally constant sheaves of $\Lambda$-modules on $\mathcal{C}$. If $\mathcal{F}$ is of finite type, then there exists a covering $\{ U_ i \to X\}$ such that $\varphi |_{U_ i}$ is the map of constant sheaves of $\Lambda$-modules associated to a map of $\Lambda$-modules.

Proof. Proof omitted. $\square$

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