Today I was thinking about an explicit example of sheafification, namely the sheafification of the presheaf of bounded continuous functions on $\mathbb{R}$. I am working with sheafification as the sheaf of sections of the corresponding étalé space.
I knew before starting that this sheafification is isomorphic to the sheaf of continuous functions. For illustrative purposes I checked this by manually constructing the sheaf of sections for the étalé space, then showing it is isomorphic to the sheaf of continuous functions and also by showing that the sheaf of continuous functions satisfies the universal property of the sheafification. After seeing this formally, I decided this is intuitive because locally, germs on the sheaf of continuous functions should look the same as the germs on the presheaf of continuous and bounded functions, and when we sheafify we just add in things that can be made from germs that were missing. What I am wondering now is if this is a coherent way to predict or intuit what a sheafification should look like. Namely, I am wondering
Is it true that for any presheaf $\mathcal{F}$ and its sheafification $\mathcal{F}^{+}$, $\mathcal{F}_x \cong \mathcal{F}^{+}_{x}$ for all $x$?
Is it true that if $\mathcal{F}$ is a presheaf and $\mathcal{G}$ is a sheaf such that $\mathcal{F}_x \cong \mathcal{G}_x$ for all $x$, then $\mathcal{G} \cong \mathcal{F}^{+}$
I am aware I am only thinking about seperated presheaves, which could be a huge flaw in my reasoning.
Edit: I suppose what I am trying to do is find a provable formalization that expresses my intuition that “if a sheaf is a presheaf who’s sections are determined by local data, then if a presheaf and a sheaf share the same most-local data (germs/stalks) and also have the same criterion for compatibility, the sheafification of the presheaf should coincide with the sheaf”