= Solution
Write $|\phi_A\rangle=\sum_jc_j|j\rangle_A$ and choose the <conjugate index vector>
$$
|\phi_B^*\rangle=\sum_jc_j^*|j\rangle_B,\qquad\langle\phi_B^*|=\sum_jc_j\langle j|_B.
$$
With $|\widetilde\Psi_{AB}\rangle=\sqrt d|\Psi_{AB}\rangle=\sum_j|j\rangle_A|j\rangle_B$, the partial inner product gives
$$
\boxed{(I_A\otimes\langle\phi_B^*|)|\widetilde\Psi_{AB}\rangle=\sum_jc_j|j\rangle_A=|\phi_A\rangle.}
$$
This is the <relative state> construction. The coefficient conjugation is defined relative to the fixed paired bases; it is essential, because the bra conjugates those coefficients again. Using the normalized entangled vector instead would give $|\phi_A\rangle/\sqrt d$.
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