I'm in the beginning stages of this proof. My question here I guess is that by definition a monoid has the properties that it is associative and has an identity. So: $(ab)c=a(bc)$ and $de=ed=e$ where $e$ is the identity. If I can prove that the identity is unique, does that prove the inverse is unique?
Of course the identity is unique, $e_1=e_1\circ e_2=e_2$. Now suppose $a,b$ are both inverses of $x$. Then
$$a = a\circ e = a\circ(x\circ b) = (a\circ x)\circ b = e\circ b = b.$$
The fact that the identity is unique plays no role, really, though the proof uses the same method of switching between the left and right perspectives of an expression and substituting with identity.