One thing we need is another fact about the tensor product of abelian groups. If we take three abelian groups , , and , we can form the tensor product , and then use that to make . On the other hand, we could have started with and then built . If we look at the construction we used to show that tensor products actually exist we see that these two groups are not the same. However, they are isomorphic.
To see this, let’s make a bilinear function from to . By our construction, any element of can be represented as a sum , so linearity says we just need to consider elements of the form . Define . This induces a unique linear function given by and extending to sums of such elements. Similarly we get a linear function , so we have an isomorphism of abelian groups. We can thus (somewhat) unambiguously talk about “the” tensor product .
Now let’s take a collection of abelian groups with running over an index set , and let be any other abelian group. We want to consider the tensor product
Since the direct sum is a direct product of groups, it comes with projections . Since the free product is in general a subgroup of the direct sum (a proper subgroup when the index set is infinite), we also have injections coming from the free product. We can use these to build homomorphisms
applying to and the identity to . By the universal property of direct sums (the one it gets from free products of groups) this gives us a homomorphism
On the other hand, for each we have a bilinear function sending in to in . By the universal properties of tensor products this gives a linear function . The universal property of direct sums (the one it gets from direct products of groups) gives us a linear function
Now there’s a lot of juggling of functions and injections and projections here that I really don’t think is very illuminating. The upshot is that and are inverses of each other, giving us an isomorphism of the two abelian groups. There’s nothing really special about the left side of the tensor product either. A similar result holds if the direct sum is the right tensorand. We can even put them together to get the really nice isomorphism: