Coproducts in the category of Sets

As with products. Not every category has coproducts. However, does. I claim that given two sets and , their coproduct is the disjoint union of and . 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 denote the disjoint union, and then verify that it really is the coproduct.

Proof:

We have the natural injections and . Suppose we have a set and and . Define by if , and if . Notice that every is in the image of precisely one of the maps and , so this map is well defined. Now we just need to check that the diagram

commutes. I’ll leave that as an exercise.