Composite of Group Epimorphisms is Epimorphism

Theorem
Let:
 * $\struct {G_1, \odot}$
 * $\struct {G_2, *}$
 * $\struct {G_3, \oplus}$

be groups.

Let:
 * $\phi: \struct {G_1, \odot} \to \struct {G_2, *}$
 * $\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$

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.