Composite of Endomorphisms is Endomorphism

Theorem
Let: be algebraic structures.
 * $\left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right)$
 * $\left({S_2, *_1, *_2, \ldots, *_n}\right)$
 * $\left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$

Let: be endomorphisms.
 * $\phi: \left({S_1, \circ_1, \circ_2, \ldots, \circ_n}\right) \to \left({S_2, *_1, *_2, \ldots, *_n}\right)$
 * $\psi: \left({S_2, *_1, *_2, \ldots, *_n}\right) \to \left({S_3, \oplus_1, \oplus_2, \ldots, \oplus_n}\right)$

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

Proof
From Composite of Homomorphisms $\psi \circ \phi$ is a homomorphism.

From Composite of Surjections is Surjection $\psi \circ \phi$ is an surjection.

An endomorphism is an surjjective endomorphism.

Hence $\psi \circ \phi$ is an endomorphism.