Proposition 3

Let B be a topological space, let $\mathscr{B}$ be a basis of the topology of B, let $\mathscr{F}= (\mathscr{F}(U), f_{UV})$ be a presheaf on B relative to $\mathscr{B}$, let $\widetilde{\mathscr{F}}$ be the associated sheaf and let $\sigma_{\mathscr{F}}:\mathscr{F}\rightarrow \widetilde{\mathscr{F}}|\mathscr{B}$ be the canonical morphism. Given a sheaf $\mathscr{G}= (\mathscr{G}(U), g_{UV})$ on B and a morphism of presheaves $\varphi :\mathscr{F}\rightarrow \mathscr{G}|\mathscr{B}$, there exists a unique morphism of sheaves $\psi :\widetilde{\mathscr{F}}\rightarrow \mathscr{G}$ such that $\psi |\mathscr{B}\circ \sigma_{\mathscr{F}}=\varphi$.