Component Mappings of Set Coproduct are Injective

Theorem
Let $S_1$ and $S_2$ be sets.

Let $\struct {C, i_1, i_2}$ be a coproduct of $S_1$ and $S_2$.

Then $i_1$ and $i_2$ are injections.

Proof
By definition of coproduct:


 * for all sets $X$ and mappings $f_1: S_1 \to X$ and $f_1: S_1 \to X$
 * there exists a unique mapping $h: C \to X$ such that:
 * $h \circ i_1 = f_1$
 * $h \circ i_2 = f_2$