I want to say that a group object in a category (e.g. a discrete group, topological group, algebraic group...) is the image under a product-preserving functor of the "group object diagram", $D$. One problem with this idea is that this diagram $D$ as a category on its own doesn't have enough structure to make the object labelled $``G\times G"$ really the product of $G$ with itself in $D$.

Is there a category $U$ with a group object $G$ in it such that every group object in every other category $C$ is the image of $G$ under a product-preserving functor $F:U\rightarrow C$, unique up to natural isomorphism?

(It's okay with me if "product-preserving" or "up to natural isomorphism" are replaced by some other appropriate qualifiers, like "limit preserving"...)