**§4. Uniqueness of the tensor product**

Starting with the usual setup, suppose and are two pairs satisfying the universal property. We can give a slick proof of uniqueness as follows: observe that due to the universal property of , the bilinear map induces a linear map such that . Similarly, due to the universal property of , the bilinear map induces a linear map such that .

Now , and , and by applying (T1) twice, we see that the vectors and generate and respectively, so therefore and are inverse linear isomorphisms and we’re done.

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