What would you like to understand?

The tensor product is unique up to isomorphism because if $(W, \varphi)$ and $(W', \varphi')$ are two tensor products of $U$ and $V$, there exists a unique linear space isomorphism $\tau: W \to W'$ such that the diagram commutes, meaning $W' = \tau(W)$ and $\varphi' = \tau \varphi$. This shows that any two realizations differ only by a unique isomorphism.

Conditions: $U$ and $V$ are linear spaces over the number field $\mathbb{P}$.; $(W,\varphi)$ is a tensor product of $U$ and $V$.; $(W',\varphi')$ is another tensor product of $U$ and $V$.

The tensor product is unique up to isomorphism because if $(W, \varphi)$ and $(W', \varphi')$ are two tensor products of $U$ and $V$, there exists a unique linear space isomorphism $\tau: W \to W'$ such that the diagram commutes, meaning $W' = \tau(W)$ and $\varphi' = \tau \varphi$. This shows that any two realizations differ only by a unique isomorphism.

Conditions: $U$ and $V$ are linear spaces over the number field $\mathbb{P}$.; $(W,\varphi)$ is a tensor product of $U$ and $V$.; $(W',\varphi')$ is another tensor product of $U$ and $V$.