quantum mechanics - Invariance of a maximally entangled state under unitary operation $U otimes U^dagger$

Apparently the (d-dimensional) maximally entangled state, $|E \rangle = \sum_{i} |ii\rangle /\sqrt{d}$ is invariant under operations of the form $U \otimes U^\dagger$. I want to prove this result, which amounts to showing that

$$\left(\sum_i U | i \rangle \otimes V |i \rangle = \sum_i | ii \rangle \right)\Rightarrow (V = U^\dagger )$$

I have no idea how to even start. I suppose it's some simple linear algebra result, but I don't see it. A hint would be appreciated.

Answer

Write the terms $U |i \rangle$ and $V | i \rangle$ explicitly in the matrix form, e.g., $U | i \rangle = \sum_{k} | k \rangle U_{ki}$ and analogously. Then play with exchanging the three summations of indices, you should get it proved.