Sheafification

Sheafification associates a sheaf to a presheaf without changing its stalks and is universal among morphisms from that presheaf to sheaves.

Every presheaf morphism from $\mathcal{F}$ to a sheaf $\mathcal{H}$ factors uniquely through the canonical sheafification map. Local representatives give local images in $\mathcal{H}$, and the sheaf gluing axiom makes their union unique.

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.