TITLE: Is there a "universal group object"? (answered: yes!)
QUESTION [21 upvotes]: 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"...)

REPLY [2 votes]: I would really suggest looking at Steve Awodey's lecture notes on Categorical Logic., found here http://www.andrew.cmu.edu/user/awodey/catlog/notes/  The category you are looking for is called "the theory of groups".  I find these notes much more digestible than Lawvere's original papers on the subject. 
Essentially, you can form a category in which all objects are products of a single object G, and the only morphisms between them are those morphisms you get out of the basic definition of a group object.  Then a group in any category is a product preserving functor from this category.  Actually, in this case, it is easy to see that the appropriate category is just the opposite of the full subcategory 1, F(1), F(2), ... , F(n), ... where F(k) is the free group on k generators.