Let $A$ be a commutative ring and $M,N,P$ be A-modules.I know that tensor product of $M$ and $N$ is a universal object ($ M \otimes N$,u) (where $M \otimes N$ is a $A$-module and $u: M\times N \to M\otimes N$ is a bilinear map) with the following universal property:
Universal Property: Given any bilinear map $ \phi: M\times N \to P$ there exist a unique Linear map $f: M\otimes N \to P$ such that $\phi=f o u$.
I've read the construction of tensor product but I don't really see the motivation for the definition of tensor product.I mean why do we need such an object? Why do we define tensor product with the above universal property.What was the Motivation for such definition of tensor product?