sheaf valued functors: how much can you prove about them just using category theory

Question

This question is related to my earlier question. I have learnt a lot since I asked that question and I think I can phrase my thoughts more clearly now.

To someone who is comfortable with category theory, it seems reasonable to define sheafification as the left adjoint to the forgetful functor from sheaves to presheaves (assuming such a functor exists). Intuitively, sheafification associates to the presheaf $\mathcal{F}$, a sheaf $\mathcal{F}^{+}$ which has isomorphic stalks. However, this is not obvious from the definition of sheafification as an adjoint.

Question: Is it possible to prove that the unit associated to the adjunction (sheafification, forgetful) induces isomorphisms on the stalks without explicitly constructing sheafification?

Note that I am not saying that defining sheafification as an adjoint is a good idea. Indeed, the explicit construction of the sheafification of a presheaf is a good exercise, but the fact that the unit $\mathcal{F} \to \mathcal{F}^+$ induces isomorphisms on the stalks seems like it should follow from properties of adjoint functors.

I have similar questions about the inverse image functor but I think that if i can find out how to solve the above question, then I should be able to work out the details for the inverse image functor aswell

Answer 1

The stalk of a (pre)sheaf $\mathcal{F}$ at a point $x$ is the colimit of the directed system of (pre)sheaves $\mathcal{F}_U = j_* j^* \mathcal{F}$, where $U$ runs over all open sets containing $x$ and $j$ is the inclusion map of $U$. (This is the sheaf version of the definition of a stalk; it realizes the stalk as a skyscraper sheaf concentrated at $x$.) Since sheafification is a left adjoint, it is right exact, i.e. preserves colimits.