Morphism of sheaves ($\varphi :\mathcal{F}\to \mathcal{G}$)

A morphism of sheaves assigns a map $\mathcal{F}(U)\to \mathcal{G}(U)$ on every open set, commuting with all restriction maps. For sheaves of abelian groups, sheaves of rings or sheaves of modules, the maps must also preserve the relevant algebraic structure. It induces maps on every stalk, and equality of sheaf morphisms can be checked there.

For a morphism of sheaves of abelian groups or modules, the kernel sheaf has sections $\ker (\mathcal{F}(U)\to \mathcal{G}(U))$. These already satisfy the sheaf gluing axiom. Its stalk at $x$ is the kernel of ${\mathcal{F}}_x\to {\mathcal{G}}_x$, since filtered colimits of modules preserve exactness. A sheaf morphism is injective exactly when its kernel sheaf is zero.