For simplicity, I'll stick to the case where you are just taking the tensor product of two spaces and . It's usual to write as , and it is part of the definition that the tensor product should be linear in each of its factors. So we need it to be true that (with a similar property for the second coordinate). If you just take the free vector space over V then these properties will not hold. So you need to quotient F(V) by the subspace M generated by all these relations.