Composite of Group Isomorphisms is Isomorphism

Theorem
Let $\struct {G_1, \circ}$, $\struct {G_2, *}$ and $\struct {G_3, \oplus}$ be groups.

Let $\phi: \struct {G_1, \circ} \to \struct {G_2, *}$ and $\psi: \struct {G_2, *} \to \struct {G_3, \oplus}$ 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 is Homomorphism, $\psi \circ \phi$ is a group homomorphism.

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