Defining multiplication on the tensor product of $R$-algebras. If $M$ and $N$ are $R$-algebras, then one can define a multiplication of elementary tensors as follows; $(m \otimes n) \cdot (m' \otimes n') = mm' \otimes nn'$. 
My question is how can we show, using the universal property of the tensor product, that this is a well defined operation?
My understanding is that we would be need a 4-linear map out of $(M \times N)\times (M \times N)$ into $M \otimes N$, that would factor through $(M \otimes N) \otimes (M \otimes N)$ giving us the required linear map. 
Am I on the right track? I can't seem to get my head around the details in the proof. Any pointers would be of great help to me.

This is explained in the first paragraph of Bourbaki, A.III.4.1.
