Restriction of Homomorphism to Image is Epimorphism
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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 $\Img \phi$.
Proof
Let $\phi: S \to T$ be a homomorphism.
Let $\Img \phi = T'$
By Morphism Property Preserves Closure, $T'$ is closed.
From Restriction of Mapping to Image is Surjection, $\phi \to \Img \phi$ 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.
$\blacksquare$