Monoid homomorphism

Definition

Let $A$ and $B$ be monoids. A mapping $f:A\to B$ is called a homomorphism if the following conditions are satisfied:

1. identity is respected: $f(1_A)=f(1_B)$, and

2. multiplication is respected: $f(ab)=(fa)(fb)$.

Monoids and homomorphisms between them form a category.