The Stacks project

Proof. Let $\mathcal{I} \to \textit{PAb}(\mathcal{C})$, $i \mapsto \mathcal{F}_ i$ be a diagram. We can simply define abelian presheaves $L$ and $C$ by the rules

and

It is clear that there are maps of abelian presheaves $L \to \mathcal{F}_ i$ and $\mathcal{F}_ i \to C$, by using the corresponding maps on groups of sections over each $U$. It is straightforward to check that $L$ and $C$ endowed with these maps are the limit and colimit of the diagram in $\textit{PAb}(\mathcal{C})$. This proves (1) and (2). Details omitted. $\square$