# Definition of stalk as a colimit of sheaves

I'm trying to understand the category-theoretic proof that sheafification preserves stalks, using adjoints, as outlined e.g. in this mathoverflow answer:

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.
Right. But the obvious way of turning the filtered colimit of sections into a filtered colimit of sheaves is by the procedure you mention. The (pre)sheaf $j_* j^* F$ can be explicitly described as the (pre)sheaf where $(j_* j^* F)(V) = F(U \cap V)$. – Zhen Lin Jul 15 '12 at 18:03
Ah, ok, this is making sense now. I was always thinking of $j_* j^*F(V)$ as some awkward colimit, but in fact it simplifies like you say to $F(U \cap V)$. From there I was able to convince myself that the colimit of sheaves coincides with the skyscraper sheaf desired. Thanks! If you wish to make your comment an answer, I'll happily accept it. – KristianJS Jul 15 '12 at 21:45