Construction of Inverse Completion/Quotient Mapping to Image is Isomorphism

Theorem
Let the mapping $\psi: S \to T\,'$ be defined as:
 * $\forall x \in S: \psi \left({x}\right) = \left[\!\left[{\left({x \circ a, a}\right)}\right]\!\right]_\boxtimes$

Let $S\,'$ be the image $\psi \left({S}\right)$ of $S$.

Then $\psi$ is an isomorphism from $S$ onto $S\,'$.

Proof
From Quotient Mapping is Monomorphism, $\psi: \left({S, \circ}\right) \to \left({S\,', \oplus'}\right)$ is a monomorphism.

Therefore by definition:
 * $\psi$ is a homomorphism
 * $\psi$ is an injection.

Because $S\,'$ is the image of $\psi \left({S}\right)$, by Surjection by Restriction of Codomain $\psi$ is a surjection.

Therefore by definition $\psi: S \to S\,'$ is a bijection.

A bijective homomorphism is an isomorphism.