Definition 2

A magma with an identity element is called a unital magma. If $E, E'$ are unital magmas, a homomorphism of the magma $E$ into the magma $E'$ which maps the identity element of $E$ to the identity element of $E'$ is called a unital homomorphism (or morphism) of $E$ into $E'$. An associative unital magma is called a monoid.

If $E, E'$ are monoids, a unital morphism of $E$ into $E'$ is called a monoid homomorphism or a monoid morphism of $E$ into $E'$.