Here's one way to answer your question. Consider the category $\mathbf{PSh}(X)$ of presheaves (of sets) on a topological space $X$. A map $F\to G$ of $\mathbf{PSh}(X)$ is said to be a local isomorphism if for every point $x\in X$, the induced map $F_x\to G_x$ on stalks is a bijection. Denote by $W$ the class of local isomorphisms. Now the category $\mathbf{Sh}(X)$ of sheaves on $X$ is equivalent to the localization $W^{-1}\mathbf{PSh}(X)$. In other words, for any presheaf $F$, there is a local isomorphism $F\to F'$, where $F'$ is a sheaf.

Here's one way to answer your question. Consider the category $\mathbf{PSh}(X)$ of presheaves (of sets) on a topological space $X$. A map $F\to G$ of $\mathbf{PSh}(X)$ is said to be a local isomorphism if for every point $x\in X$, the induced map $F_x\to G_x$ on stalks is a bijection. Denote by $W$ the class of local isomorphisms. Now the category $\mathbf{Sh}(X)$ of sheaves on $X$ is equivalent to the localization $W^{-1}\mathbf{PSh}(X)$. In particular, for any presheaf $F$, there is a local isomorphism $F\to F'$, where $F'$ is a sheaf.