(i) | $1_G$ and $1_H$ are each an D39: Identity element in $G$ and $H$, respectively |

**monoid homomorphism**from $G$ to $H$ if and only if

(1) | \begin{equation} \forall \, x, y \in G : f(x y) = f(x) f(y) \end{equation} |

(2) | \begin{equation} f(1_G) = 1_H \end{equation} |