What is the "object" of a group (when viewed as a category) analogous to in a group (when viewed as a group)?

Question

From MacLane's Category Theory:

A group is a category with one object in which every arrow has a two-sided inverse under composition.

So the analogies I see between a category group $CG$ and a group $G$ is as follows:

Arrows : Elements of the group

Identity arrow : identity element

composition of arrows : group operation

Associativity function : associative group operation

Unit law : identity properties

All composable arrows exist : closure of the group

But the one I can't seem to understand is: What is analog for the single object in $CG$?

Single object in category : (Something for a group)?

Answer 1

The single object in this category could be the set of elements in the group, where each morphism corresponds to the function from the group to itself given by multiplying by an element of the group. That way both $\hom(G, G)$ and $G$ itself correspond to the elements of the group, but in different ways. That's the most natural thing that I can think of.

But at the end of the day, it's not really important exactly what this object is.

Answer 2

While Arthur’s answer is correct, in most constructions the object isn’t anything. It’s just a formality introduced to give us somewhere to stick the morphisms. It’s a standard point in category theory that the objects aren’t where the content is, and that’s particularly true here.