The Stacks project

Lemma 6.16.4. let $X$ be a topological space. Let $(\mathcal{C}, F)$ be a type of algebraic structure. Suppose that $\mathcal{F}$, $\mathcal{G}$ are sheaves on $X$ with values in $\mathcal{C}$. Let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of the underlying sheaves of sets. If for all points $x \in X$ the map $\mathcal{F}_ x \to \mathcal{G}_ x$ is a morphism of algebraic structures, then $\varphi $ is a morphism of sheaves of algebraic structures.

Proof. Let $U$ be an open subset of $X$. Consider the diagram of (underlying) sets

\[ \xymatrix{ \mathcal{F}(U) \ar[r] \ar[d] & \prod _{x \in U} \mathcal{F}_ x \ar[d] \\ \mathcal{G}(U) \ar[r] & \prod _{x \in U} \mathcal{G}_ x } \]

By assumption, and previous results, all but the left vertical arrow are morphisms of algebraic structures. In addition the bottom horizontal arrow is injective, see Lemma 6.11.1. Hence we conclude by Lemma 6.15.4, see also Example 6.15.5 $\square$

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