tensor product of algebras.
Ok, so you have the map by . I agree that it's the right map. So, recall that an isomorphism of associative unital algebras is a linear isomorphism which is multiplicative and maps the multiplicative identity to the multiplicative identity (i.e. it preserves both the vector space and monoidal structures). It's clearly linear because you defined it to be so. In particular, you've defined (really you've defined it on more than a basis, but that's unimportant) on a basis of and of course are letting be the unique linear map. So the question is why it respects the multiplicative properties. It clearly maps the identity to the identity since the identity for is where is the identity map on and thus for any we have that so that . Can you do the multiplicative part?