Global sections of an abelian sheaf

Fix a topological space $X$ and write $\mathrm{A}\mathrm{b}(X)$ for the category of sheaves of abelian groups on $X$ (Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories). For a sheaf $\mathcal{F}$ on $X$ put $\mathrm{\Gamma }(X,\mathcal{F}):=\mathcal{F}(X),$ the abelian group of global sections of $\mathcal{F}$ (Sections, restrictions, and global sections of a presheaf), and for a morphism $\phi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves put $\mathrm{\Gamma }(X,\phi ):={\phi }_X:\mathcal{F}(X)⟶\mathcal{G}(X)\mathrm{.}$ Since a morphism of sheaves is a morphism of the underlying presheaves, it is a family of group homomorphisms commuting with restriction (Morphisms of presheaves), and composites and identities are computed componentwise; hence $\mathcal{F}\mapsto \mathrm{\Gamma }(X,\mathcal{F})$ and $\phi \mapsto \mathrm{\Gamma }(X,\phi )$ define the global-sections functor $\mathrm{\Gamma }(X,-):\mathrm{A}\mathrm{b}(X)\to \mathbf{A}\mathbf{b}$.

Addition of morphisms $\mathcal{F}\to \mathcal{G}$ is componentwise (Sheaves of abelian groups, and likewise sheaves of modules on a ringed space, form abelian categories), so $\mathrm{\Gamma }(X,\phi +\psi )=\mathrm{\Gamma }(X,\phi )+\mathrm{\Gamma }(X,\psi )$ and $\mathrm{\Gamma }(X,-)$ is additive. It is left exact: if $0\to {\mathcal{F}}^{\mathrm{\prime }}\to \mathcal{F}\to {\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }}$ is exact in $\mathrm{A}\mathrm{b}(X)$, then $0\to \mathrm{\Gamma }(X,{\mathcal{F}}^{\mathrm{\prime }})\to \mathrm{\Gamma }(X,\mathcal{F})\to \mathrm{\Gamma }(X,{\mathcal{F}}^{\mathrm{\prime }\mathrm{\prime }})$ is exact (Global sections are left exact but need not preserve epimorphisms). It is not exact in general, since it need not preserve epimorphisms (Global sections are left exact but need not preserve epimorphisms).