This is a follow up question to my previous question here.
I'm confused about the following: in Atiyah-Macdonald they state that there exists a unique isomorphism $M \otimes N \to N \otimes M$, $m \otimes n \mapsto n \otimes m$.
I'm not sure why AM write unique: If I already know the map, $m \otimes n \mapsto n \otimes m$, then there is no other map that is exactly the same. So I think I get uniqueness for free and all I have to show is that $m \otimes n \mapsto n \otimes m$ is an isomorphism.
On the other hand, and that's the way I understood the proposition in the book, if I want to show that there exists a unique isomorphism (without knowing what it is in advance) then I can use the universal property of the tensor product to get uniqueness and the map will turn out to be $m \otimes n \mapsto n \otimes m$ which is an isomorphism.
What am I missing?