monoid homomorphism

Let \(M,N\) be monoids. A monoid homomorphism is a map

\begin{align*} \varphi: M \rightarrow N \\ m \mapsto \varphi(m) \end{align*}

such that

1. \(\varphi(e_m) = e_n\), see neutral element
2. \(\varphi(m_1 \cdot m_2) = \varphi(m_1) \cdot \varphi(m_2)\)