I am new to tensors and got stuck with a theorem related to Tensor contractions. It says given a map $Z$ from ordered pair of Cartesian Product to a Tensor space, we are sure that there exists another map $C$ from the Tensor Product of the same underlying vector space to the tensor product space. This is called "Universal Property".
Can anybody help me with the proof that such a map exists and what that map is?
Given, $Z: V \times V \to U$. Then, $C: V \otimes V \to U$ exists. [$\otimes$ signifies tensor product]
Thank you.