Composite of Group Isomorphisms is Isomorphism

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

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

Then the composite of $\phi$ and $\psi$ is also a group isomorphism.

Proof
A group isomorphism is a group homomorphism which is also a bijection.

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

From Composite of Bijections is Bijection, $\psi \circ \phi$ is a bijection.