Ken Scambler wrote:

>The inverse operation takes x to -x, -x to x, and the identity 0 to itself. The group equations follow: -x + x = x + -x = 0 and 0 = -0

I don't think that will work since it just swaps inversion.

However, \$$-x \mapsto x\$$, \$$x \mapsto x\$$, and \$$0 \mapsto 0\$$ I think will work.

The adjoint here takes a monoid and freely adds inverse elements, while it's right adjoint forgets what it means to be an inverse.

In the context of numbers, the adjoint is the multiplication of a natural number by \$$-1\$$, \$-n : \mathbb{N} \to \mathbb{Z},\$ it's right adjoint is the absolute value, \$|z|: \mathbb{Z} \to \mathbb{N}.\$