I'll have a stab at 164:
It has a left adjoint that generates an inverse element -x for every non-identity element x.
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
This is like how integers can be freely generated from natural numbers.
I'll guess it has a right adjoint by analogy to the German/Italian example where the right adjoint threw away all the information into a crappy trivial mapping onto a single Italian. So I guess we melt the monoid down into the* trivial group with one element, where everything maps to id.
Left = generous, right = stingy!