Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories

In both categories, sums and zero morphisms are defined sectionwise, the zero sheaf is a zero object, and finite direct sums are obtained by taking direct sums on every open set. Hence both categories are additive.

By [F2], every morphism has a kernel and a cokernel. The same item also gives the image sheaf by sheafifying the objectwise image presheaf, and the coimage is the cokernel of the kernel inclusion.

Let $\phi :\mathcal{F}\to \mathcal{G}$ be a morphism in either category. At a point $x\in X$, the stalks of the kernel, cokernel, image, and coimage from step 2.1 are the usual kernel, cokernel, image, and coimage of the stalk map ${\phi }_x$, because kernels are objectwise and [L2] identifies stalks after sheafification. By [L1], module categories are abelian, and the abelian-group case is the special case of modules over $\mathbb{Z}$, so the canonical map $\mathrm{coim}({\phi }_x)\to \mathrm{im}({\phi }_x)$ is an isomorphism for every $x$.

The sheaf morphism $\mathrm{coim}(\phi )\to \mathrm{im}(\phi )$ is therefore an isomorphism on every stalk, so [L3] makes it an isomorphism globally. Together with steps 1.1 and 2.1, [F1] shows that both categories are abelian.