Let $\mathbf{M_1}=(M_1,\star,e)$ and $\mathbf{M_2}=(M_2, \square, \epsilon)$ are two monoides. If $h: \mathbf{M_1} \rightarrow \mathbf{M_1} \times \mathbf{M_2}$ and $h(a)=(a,\epsilon)$ prove:
a) $h$ is homomorphism and $\text{1-1}$ of monoid $\mathbf{M_1}$ to $\mathbf{M_1} \times \mathbf{M_2}$
b) $h$ with inherited operations follows $h(\mathbf{M_1}) \cong \mathbf{M_1}$.
I have the problem. I do not know how to show homomorphism between two monoides because there is no definition of operation $\star$ and $\square$, and that problem was on my exam yesterday. I know for homomorphism must be $h(a \star b) = \ ? \ = h(a) \ \square \ h(b)$. I appreciate your help.