I want to say that a [group object](http://en.wikipedia.org/wiki/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 $D$.



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



Is there a "universal group object"? (answered: yes!)
Yes, the category U is the opposite of the full subcategory of Grp on the free groups on 0, 1, 2, ... generators. This is an instance of Lawvere's theory of "theories". See [this nLab entry](http://ncatlab.org/nlab/show/Lawvere+theory) for a discussion (of this example in fact).


