Existence of Bijection between Coproducts of two Sets

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

Let $\struct {C, i_1, i_2}$ and $\struct {D, j_1, j_2}$ be two coproducts on $S_1$ and $S_2$.

Then there exists a unique bijection $\theta: D \to C$ such that:
 * $\theta \circ j_i = i_1$
 * $\theta \circ j_2 = i_2$

Proof
Let $X$ be an arbitrary set.

Let $f_1: S_1 \to X$ and $f_2: S_2 \to X$ be arbitrary mappings.

Let $h_C: C \to X$ be the unique mapping such that:
 * $h_C \circ i_1 = f_1$
 * $h_C \circ i_2 = f_2$

Let $h_D: D \to X$ be the unique mapping such that:
 * $h_D \circ j_1 = f_1$
 * $h_D \circ j_2 = f_2$

The existence and uniqueness of $h_C$ and $h_D$ follow from the fact of $\struct {C, i_1, i_2}$ and $\struct {D, j_1, j_2}$ both being coproducts of $S_1$ and $S_2$.

We have that:
 * $h_C \circ i_1 = f_1 = h_D \circ j_1$
 * $h_C \circ i_2 = f_2 = h_D \circ j_2$