I'm trying to understand the category-theoretic proof that sheafification preserves stalks, using adjoints, as outlined e.g. in this mathoverflow answer:
What is bugging me is that I'm having a hard time seeing why the colimit described is actually isomorphic to the stalk (as a skyscraper sheaf based at x using the stalk as the fixed set).
Specifically, in his answer, Ryan Reich claims that for a sheaf $F$ on $X$, the stalk $F_x$ is the colimit over opens $U$ containing $x$ of the sheaves $j_* j^* F$, where $j: U \rightarrow X$ is the inclusion map.
Every time I try to show that the skyscraper sheaf satisfies the universal property I end up in a seeming dead end. I feel like it shouldn't be THAT hard, and that I must be missing something or confusing myself somehow. Does anyone know a reference for this, or be willing to explain why it works? I'd be very grateful.