Canonical Injection is Monomorphism

Theorem
Let $\left({S_1, \circ_1}\right)$ and $\left({S_2, \circ_2}\right)$ be algebraic structures with identities $e_1, e_2$ respectively.

The canonical injections:


 * $\operatorname{in}_1: \left({S_1, \circ_1}\right) \to \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right): \forall x \in S_1: \operatorname{in}_1 \left({x}\right) = \left({x, e_2}\right)$


 * $\operatorname{in}_2: \left({S_2, \circ_2}\right) \to \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right): \forall x \in S_2: \operatorname{in}_2 \left({x}\right) = \left({e_1, x}\right)$

are monomorphisms.

Proof
From Canonical Injections are Injections we have that the canonical injections are in fact injective.

It remains to prove the morphism property.

Let $x, y \in \left({S_1, \circ_1}\right)$.

Then:

and the morphism property has been demonstrated to hold for $\operatorname{in}_1$.

Thus $\displaystyle \operatorname{in}_1: \left({S_1, \circ_1}\right) \to \left({S_1, \circ_1}\right) \times \left({S_2, \circ_2}\right)$ has been shown to be an injective homomorphism and therefore a monomorphism.

The same argument applies to $\operatorname{in}_2$.