The definition of monoid homomorphisms requires that $\eta(1)=1'$. However, I doubt whether this condition is superfluous, for
$\forall x \in M, \eta(x) = \eta(x*1) = \eta(x)\eta(1)$
$\forall x \in M, \eta(x) = \eta(1*x) = \eta(1)\eta(x)$
so we may infer that $\eta(1)$ acts as $1'$ in $\eta(M)$, utilizing the condition $\forall a, b \in M, \eta(ab)=\eta(a)\eta(b)$ only.
So is it really necessary to emphasize this condition?