Let and be finite dimensional vector spaces, and define to be the vector space of all bilinear maps . Given a bilinear map , define by . Define a map by

In order to show that satifies the universal property of the tensor product, I have to show that given a map , then there is a unique such that , where is the canonical isomorphism.

It is quite clear that defined above satisfies this property, but I am having trouble proving uniqueness. I would like to show that given such that , then , however I am getting nowhere. Any help would be appreciated, thank you. Ideally I'd like to show that .

Note that I don't want to invoke the existence of a map , I want to verify directly that is in fact the tensor product.