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 "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, like "limit preserving"...)