General Morphism Property for Semigroups

Theorem
Let $$\left({S, \circ}\right)$$ and $$\left({T, \ast}\right)$$ be semigroups.

Let $$\phi: S \to T$$ be a homomorphism.

Then $$\forall s_k \in S: \phi \left({s_1 \circ s_2 \circ \cdots \circ s_n}\right) = \phi \left({s_1}\right) \ast \phi \left({s_2}\right) \ast \cdots \ast \phi \left({s_n}\right)$$.

Hence it follows that $$\forall n \in \mathbb{N}^*: \forall s \in S: \phi \left({s^n}\right) = \left({\phi \left({s}\right)}\right)^n$$.

Proof
$$\forall s_k \in S: \phi \left({s_1 \circ s_2 \circ \cdots \circ s_n}\right) = \phi \left({s_1}\right) \ast \phi \left({s_2}\right) \ast \cdots \ast \phi \left({s_n}\right)$$ can be proved by induction.

The result for $$n \in \mathbb{N}^*$$ follows directly from the above, by replacing each occurrence of $$s_k$$ with $$s$$.