Sheafification by locally representable germs (${\mathcal{F}}^{+}(U)\subseteq {\prod }_{x\in U}{\mathcal{F}}_x$)

For a presheaf, take families of elements of its stalks which near each point are the germs of one presheaf section. Pointwise operations and restriction make these families into a sheaf. The map sending a section to all its germs is its sheafification map.