I'm confusing about the following statement:
The category of sheaves is the localization of the category of presheaves respect to the local isomorphisms.
But what are local isomorphisms? According to the nLab entry, it seems that the following definition is correct:
A presheaf morphism is a local isomorphism iff it becomes an isomorphism after applying the sheafification.
I'm right?
If so, then how to prove the first statement?
More precisely, the sheafification $\mathscr{F}^{\sharp}$ of a presheaf $\mathscr{F}$ should satisfying \begin{equation} \mathscr{F}^{\sharp}(U) = Hom_{Sh(\mathcal{C})}(\hat{U},\mathscr{F}^{\sharp}). \end{equation} Therefore, I need to show \begin{equation} \mathscr{F}^{\sharp}(U) = \underrightarrow{\lim}_{LI_U} Hom_{PSh(\mathcal{C})}(\mathscr{U},\mathscr{F}), \end{equation} where $LI_U$ is the subcategory of $PSh(\mathcal{C})/\Upsilon(U)$ consisting of only local isomorphisms.
But how?
Obviously there is a canonical map from the colimit to $\mathscr{F}^{\sharp}(U)$ induced from the composes: \begin{equation} Hom_{PSh(\mathcal{C})}(\mathscr{U},\mathscr{F})\to Hom_{Sh(\mathcal{C})}(\mathscr{U}^{sharp},\mathscr{F}^{sharp})\to Hom_{Sh(\mathcal{C})}(\hat{U},\mathscr{F}^{\sharp}) \end{equation} where the second map is induced by the inverse of the sheafification of $\mathscr{U}\to\Upsilon(U)$.
How to show this canonical map is an isomorphism?