Universal property of sheafification ($\mathrm{Hom}({\mathcal{F}}^{+},\mathcal{H})\cong {\mathrm{Hom}}_{\mathrm{pre}}(\mathcal{F},\mathcal{H})$)

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.