Restriction of Homomorphism to Image is Epimorphism

Theorem
For any homomorphism $$\phi: S \to T$$ which is not an epimorphism, 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 Surjection by Restriction of Codomain, $$\phi \to \operatorname{Im} \left({\phi}\right)$$ 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.