Composite of Group Isomorphisms is Isomorphism

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

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

Then the composite of $\psi$ with $\phi$ 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.