It is clear why it has this property. In fact, if I have any bilinear map from M times N to P, I can also define a map from E to P Well, it sends delta_m,n to f(m,n). WELL, it sends delta_m,n to f(m,n). And now, when my map f is bilinear, this map from E to P must factor through the quotient, as f is bilinear. This map must be zero on F. So I can factor by F here, and I have such a diagram. And I also have uniqueness because my map from E to P, is to determined by images of delta_m,n. So this factorization is determined by images of delta_m,n. So this is unique. So I can call it phi and I can identify this with M otimes N. Now what is clear about my object I just introduced? That it is generated by those classes of delta_m,n modulo F. I shall denote them "m tensor n". So with these are just delta_m,n, modulo F, in fact any element of my tensor product is a finite sum of such things. Well, but of course, not equal to the set of those combinations, sorry, those tensor products, so Remark: not equal to the set of such (m tensor n)'s because E was a free module generated by those deltas, right? Well in fact, any element from the tensor product I can write as a finite sum of such symbols. But I cannot reduce these further. Now, you can ask why haven't I just defined the tensor product by this construction? Why am I talking of this universal property? And the answer is because it is easier to prove things this way. So advantages of the universal property is as follows: that the proofs become easy. For example, if I have to prove commutativity, let us prove that M tensor N, well, in fact we wright over A, if it is a tensor product of two A-modules, we write M tensor N over A, which I have not done before but I shall do this from now on. Well, if we want to prove that this is the same as N tensor M over A, then it is very elegant with the universal property. So, indeed the map from M times N to N times M which sends m, n to n tensor m is bilinear. to n tensor m is bilinear. Therefore, it factors through the tensor product. We have M tensor N to N tensor M. And to construct the inverse of alpha we do just the same. In the same way obtain the inverse map in the other direction. So, in the same way we prove, for instance, the same type of argument yields for instance, that A tensor M over A is just isomorphic to M. Well, let me prove something slightly more serious. More seriously, we have seen that the tensor product is generated by those little tensor products. So in others words, we have seen that, if M is generated by e_1, ... , e_n, and N is generated by epsilon_1, ... , epsilon_m. then the tensor product is generated by those e_i tensor epsilon_j. Well this is easy. What is less easy is to prove that now if M and N are free A-modules with basis e_1, ..., e_n that now if M and N are free A-modules with basis e_1, ..., e_n is a basis of M, and epsilon_1, ... ..., epsilon_m is the basis of N, then e_i tensor epsilon_j, where i is between 1 and n and j is between 1 and m, is a basis of M tensor N. And this is easily done with the universal property. Well, how we can do such a thing? So, I think this deserves a name. Let's call it Proposition 1. So, proof: Let us define a bilinear map from M x N to A. Sending, let's say the sum of a_i e_i, sum of b_j epsilon_j to sum of b_j epsilon_j to a_i0, b_j0. So, let me call it by a name Let me call it, say, f_i0,j0. This is bilinear. So it factors through the tensor product. And what happens, what happens, here we have, of course, that so, let me call it f-tilda i0,j0. so, let me call it f-tilda i0,j0, this f-tilda i0,j0 sends e_i0 tensor epsilon j0 to 1. sends e_i0 tensor epsilon j0 to 1. And all the other things to 0. So if I have a linear combination of my tensor products If I have the sum of alpha_ij e_i tensor epsilon_j which is equal to 0, then applying this f-tilda, we see that alpha_i0j0 is 0. we see that alpha_i0j0 is 0. But we can do this for any i0 and j0 for all i0, j0 conclude, that all coefficients are 0. So, in particular we come back to a notion, which is probably very familiar for you already, the tensor product of vector spaces. So in particular, the tensor product of vector spaces, say K-vector spaces with basis e_1 ... e_n and epsilon_1 ... epsilon_m is K-vector space with basis the e_i epsilon_j. space with basis the e_i epsilon_j. This is how it is often defined. One just introduces formally such a base and builds a vector spaces on this. But in general it is much better to use this universal property.