Coproducts in the category of Sets

As with products. Not every category has coproducts. However, \textsc{Set} does. I claim that given two sets A and B, their coproduct A\coprod B is the disjoint union of A and B. Since coproducts are unique, all we need to do is show that the disjoint union satisfies the properties of a coproduct. So we can let A\coprod B denote the disjoint union, and then verify that it really is the coproduct.

Proof:

We have the natural injections i_A:A\to A\coprod B and i_B:B\to A\coprod B. Suppose we have a set C and f_A:A\to C and f_B:B\to C. Define g:A\coprod B\to C by g(x)=f_A(x) if x\in\mbox{im}\ i_A, and g(x)=f_B(x) if x\in\mbox{im}\ i_B. Notice that every x\in A\coprod B is in the image of precisely one of the maps i_A and i_B, so this map g is well defined. Now we just need to check that the diagram

commutes. I’ll leave that as an exercise.

\square

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