Remark

Let B be a topological space, $\mathscr{F}$ a presheaf on B, $\mathscr{G}$ a sheaf on B and $\varphi :\mathscr{F}\rightarrow \mathscr{G}$ a morphism of presheaves. The canonical morphism $\sigma_{\mathscr{G}}:\mathscr{G}\rightarrow \widetilde{\mathscr{G}}$ is an isomorphism by corollary 1. By relation (2) of I, p. 54, the unique morphism $\psi :\widetilde{\mathscr{F}}\rightarrow \mathscr{G}$ such that $\psi \circ \sigma_{\mathscr{F}}=\varphi$ is therefore $\sigma^{-1}_{\mathscr{G}}\circ \widetilde{\varphi}$.