Vector space/Tensor product/Universal property/Determination/Fact

Let ${}K$ be a field, and let ${}V_{1},\ldots ,V_{n}$ denote vector spaces over ${}K$. Let ${}Z$ be a ${}K$-vector space, together with a multilinear mapping

$\rho \colon V_{1}\times \cdots \times V_{n}\longrightarrow Z,$

and suppose that ${}(Z,\rho )$ fulfills the universal property from fact  (2).

Then there exists a uniquely determined

isomorphism

$V_{1}\otimes _{}\cdots \otimes _{}V_{n}\longrightarrow Z.$

Proof, Write another proof