We first deal with the case of abelian groups as a model for the general case.
Lemma 6.12.1. Let $X$ be a topological space. Let $\mathcal{F}$ be a presheaf of abelian groups on $X$. There exists a unique structure of an abelian group on $\mathcal{F}_ x$ such that for every $U \subset X$ open, $x\in U$ the map $\mathcal{F}(U) \to \mathcal{F}_ x$ is a group homomorphism. Moreover, holds in the category of abelian groups.
\[ \mathcal{F}_ x = \mathop{\mathrm{colim}}\nolimits _{x\in U} \mathcal{F}(U) \]
Proof. We define addition of a pair of elements $(U, s)$ and $(V, t)$ as the pair $(U \cap V, s|_{U\cap V} + t|_{U \cap V})$. The rest is easy to check. $\square$
What is crucial in the proof above is that the partially ordered set of open neighbourhoods is a directed set (Categories, Definition 4.21.1). Namely, the coproduct of two abelian groups $A, B$ is the direct sum $A \oplus B$, whereas the coproduct in the category of sets is the disjoint union $A \amalg B$, showing that colimits in the category of abelian groups do not agree with colimits in the category of sets in general.
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