I try to improve my understanding of the tensor product of vector spaces. We have a universal morphism from $A\times B$ to $A\otimes B$ which is clearly not injective, but the map is "injective enough" to provide unique factorization for bilinear maps. The reason for this is, as it seems to me, that the preimages of the elementary tensors which form a basis of the tensor product space are not a basis of the product vector space. So my (probably vague) question is: When can we consider a bilinear map to be injective (I think monomorphic would be the better term but I don't understand in which category the bilinear maps fit as morphisms).
Kind regards, Sebastian