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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 $V$.

The isomorphism $\hom(U, \hom(V, W)) \cong \hom(U \otimes V, W)$ reflects the adjoint/duality relationship between Hom and tensor product. A linear map from $U$ to the space of linear maps from $V$ to $W$ corresponds to a bilinear map from $U \times V$ to $W$, which in turn corresponds to a linear map from the tensor product space $U \otimes V$ to $W$.

Conditions: $U, V, W$ are linear spaces over the number field $\mathbb{P}$.; $\hom(X,Y)$ denotes the vector space of all linear maps from $X$ to $Y$.

The isomorphism $\hom(U, \hom(V, W)) \cong \hom(U \otimes V, W)$ reflects the adjoint/duality relationship between Hom and tensor product. A linear map from $U$ to the space of linear maps from $V$ to $W$ corresponds to a bilinear map from $U \times V$ to $W$, which in turn corresponds to a linear map from the tensor product space $U \otimes V$ to $W$.

Conditions: $U, V, W$ are linear spaces over the number field $\mathbb{P}$.; $\hom(X,Y)$ denotes the vector space of all linear maps from $X$ to $Y$.