Monoid homomorphism ($h(xy)=h(x)h(y),h(1_M)=1_N$)

A monoid homomorphism $h:M\to N$ preserves multiplication and identity: $h(xy)=h(x)h(y)$ and $h(1_M)=1_N$. It therefore preserves every finite ordered product, including the empty product. The monoids with these maps form the category $\mathbf{Mon}$.