Composite of Group Epimorphisms is Epimorphism

Theorem
Let:
 * $\left({G_1, \circ}\right)$
 * $\left({G_2, *}\right)$
 * $\left({G_3, \oplus}\right)$

be groups.

Let:
 * $\phi: \left({G_1, \circ}\right) \to \left({G_2, *}\right)$
 * $\psi: \left({G_2, *}\right) \to \left({G_3, \oplus}\right)$

be (group) epimorphisms.

Then the composite of $\phi$ and $\psi$ is also a (group) epimorphism.

Proof
A group epimorphism is a group homomorphism which is also a surection.

From Composite of Group Homomorphisms is Homomorphism, $\psi \circ \phi$ is a group homomorphism.

From Composite of Surjections is Surjection, $\psi \circ \phi$ is a surection.