Construction of Inverse Completion/Quotient Mapping is Monomorphism

Theorem
Let the mapping $\psi: S \to T'$ be defined as:
 * $\forall x \in S: \map \psi x = \eqclass {\tuple {x \circ a, a} } \boxtimes$

The mapping $\psi: S \to T'$ is a monomorphism.

Proof
We have that this quotient mapping $\psi: S \to T'$ is an injection.

Let $x, y \in S$. Then:

So $\map \psi {x \circ y} = \map \psi x \oplus' \map \psi y$, and the morphism property is proven.

Thus $\psi$ is an injective homomorphism, and so by definition a monomorphism.