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 range of $$\phi$$ to its image $$\mathrm{Im} \left({\phi}\right)$$.

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

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

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

From Surjective Restriction by Limiting Range, $$\phi \to \mathrm{Im} \left({\phi}\right)$$ is a surjection.

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

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