Composite of Isomorphisms is Isomorphism/Algebraic Structure

Theorem
Let:
 * $\struct {S_1, \circ_1, \circ_2, \ldots, \circ_n}$
 * $\struct {S_2, *_1, *_2, \ldots, *_n}$
 * $\struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

be algebraic structures.

Let:
 * $\phi: \struct {S_1, \circ_1, \circ_2, \ldots, \circ_n} \to \struct {S_2, *_1, *_2, \ldots, *_n}$
 * $\psi: \struct {S_2, *_1, *_2, \ldots, *_n} \to \struct {S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}$

be isomorphisms.

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

Proof
If $\phi$ and $\psi$ are both isomorphisms, then they are by definition:
 * homomorphisms
 * bijections.

From Composite of Homomorphisms on Algebraic Structure is Homomorphism:
 * $\phi \circ \psi$ and $\psi \circ \phi$ are both homomorphisms.

From Composite of Bijections is Bijection:
 * $\phi \circ \psi$ and $\psi \circ \phi$ are both bijections.

Hence by definition both are also isomorphisms.