The title tells everything: an abelian group object in a category $\mathbf C$ with finite products is a triple $(G,m,e)$, $m\colon G\times G\to G$, $e\colon 1\to G$ such that the well known diagrams commute and such that $m\circ \sigma_{GG}=m$, where $\sigma_{AB}\colon A\times B\to B\times A$ is the map turning $\mathbf C$ into a symmetric monoidal cat.
Thanks in advance!