Let $\mathcal F$ be a sheaf on a topological space $X$, i.e. a functor $\mathcal F: \mathbf{Ouv}(X)\to R$ with some algebraic category $R$, for example rings. The functor $\mathcal F$ assigns an algebraic structure to each open set of $X$.
For any point $x$ we get a set of neighbourhoods $\{U_i\}$ ($x\in U_i$ for all $i\in I$), which forms a directed system by inclusion: we have a "direction" $U_i \to U_j$ iff $U_i\subset U_j$.
As I understand it, the concept of a stalk $\mathcal F_x$ of $x\in X$ tries to capture the sheaf's local behavior around a point $x\in X$. To realize this, I would expect to have inclusions $\mathcal F_x\subset U_i$ for each $i\in I$. The stalk $\mathcal F_x$ should be something very small around $x$.
Now, the stalk $\mathcal F_x$ of $x$ is defined as the direct limit of the directed system $\{U_i\}$, which is the colimit of the directed system as a functor. This means that instead of the inclusions $\mathcal F_x\subset U_i$, we have arrows (directions, i.e. inclusions) $U_i\to\mathcal F_x$, i.e. $U_i\subset\mathcal F_x$.
I think that cannot be correct. I really expect an inclusion $\mathcal F_x\subset U_i$, and not the other way round.
What did I understand wrong? Are the two linked Wiki pages correct?