Corollary 1

Let B be a topological space, $\mathscr{F}$ a presheaf on B$,\widetilde{\mathscr{F}}$ the associated sheaf and $\sigma_{\mathscr{F}}:\mathscr{F}\rightarrow \widetilde{\mathscr{F}}$ the canonical morphism. For $\mathscr{F}$ to be a sheaf, it is necessary and sufficient that $\sigma_{\mathscr{F}}$ be an isomorphism.

If $\sigma_{\mathscr{F}}$ is an isomorphism, $\mathscr{F}$ is a sheaf. Conversely, if $\mathscr{F}$ is a sheaf, there exists by proposition 3 a morphism $\varphi :\widetilde{\mathscr{F}}\rightarrow \mathscr{F}$ such that $\varphi \circ \sigma_{\mathscr{F}}=$ Id$_{\mathscr{F}}$. Since Id$_{\mathscr{F}}$ is the unique morphism $\psi :\widetilde{\mathscr{F}}\rightarrow \widetilde{\mathscr{F}}$ such that $\psi \circ \sigma_{\mathscr{F}}=\sigma_{\mathscr{F}}$, one has therefore $\widetilde{\sigma}_{\mathscr{F}}\circ \varphi =$ Id$_{\widetilde{\mathscr{F}}}$.