Restriction of Homomorphism to Image is Epimorphism

Theorem
Let $S$ and $T$ be algebraic structures.

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

Then a surjective restriction of $\phi$ can be produced by limiting the codomain of $\phi$ to its image $\operatorname{Im} \left({\phi}\right)$.

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

Let $\operatorname{Im} \left({\phi}\right) = T'$

By Morphism Property Preserves Closure, $T'$ is closed.

From Restriction of Mapping to Image is Surjection, $\phi \to \operatorname{Im} \left({\phi}\right)$ is a surjection.

Thus $\phi: S \to T$ is an epimorphism.

Therefore, by suitably restricting the codomain of a homomorphism, it is possible to regard it as an epimorphism.