Proof that sheafification induces isomorphism on stalks using adjoints

Question

Let $\mathcal{F}$ be a presheaf on some topological space $X$. It is not hard to prove directly that the map $\mathcal{F}\rightarrow \mathcal{F}^{sh}$ induces an isomorphism of stalks (Here $\mathcal{F}^{sh}$ is the sheafification of $\mathcal{F}$).

I want to improve on using adjoints to simplify arguments, so I was hoping someone has a proof of this using adjoints. Perhaps we can use the fact that sheafification is left-adjoint to the forgetful functor, or that the inverse image functor is left adjoint to the push forward functor?

Answer 1

This is a consequence of two facts:

• Adjoints are unique up to unique isomorphism.
• The adjoint of a composite is the composite of the adjoints.

The key observation is that the functor that sends a presheaf to its stalk has the "same" right adjoint as the functor that sends a sheaf to its stalk, namely the skyscraper (pre)sheaf functor.