= Solution
Extend the normalized <vector> $|\phi\rangle$ to an <orthonormal basis> $\{|f_j\rangle\}_{j=1}^d$ with $|f_1\rangle=|\phi\rangle$, and extend $|\psi\rangle$ to an <orthonormal basis> $\{|e_j\rangle\}_{j=1}^d$ with $|e_1\rangle=|\psi\rangle$. The <Gram-Schmidt process> provides such extensions. Define
$$
U=\sum_{j=1}^d|e_j\rangle\langle f_j|.
$$
Then $U^\dagger U=\sum_j|f_j\rangle\langle f_j|=I$ and $UU^\dagger=I$, so $U$ is unitary and $U|\phi\rangle=|\psi\rangle$. Construct $V$ on the second subsystem in the same way, sending $|\phi'\rangle$ to $|\psi'\rangle$. Consequently
$$
(U\otimes V)(|\phi\rangle\otimes|\phi'\rangle)=|\psi\rangle\otimes|\psi'\rangle.
$$
Thus \b[any two normalized <product states> on the same pair of subsystem spaces are <local-unitary equivalent>]. The dimension and <normalization> requirements matter because a unitary preserves both its domain and <vector> <norms>.
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