Restriction of Homomorphism is Homomorphism

Theorem
Let $\struct {S, \circ}$ and $\struct {T, \odot}$ be algebraic structures.

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

Let $A \subseteq S$ be a subset of $S$.

Then the restriction of $\phi$ to $A \times \Img A$ is also a homomorphism.