I want to say that a [group object][1] 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 "GxG" 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->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.)


  [1]: http://en.wikipedia.org/wiki/Group_object