In Nielsen's book when proving "Unitary freedom in the ensemble for density matrices"(Theorem 2.6):
$$\text{Suppose }|\widetilde{\psi_i}\rangle = \sum\limits_{j}u_{ij} |\widetilde{\phi_j}\rangle$$ Then in Equation 2.168: $$ \sum_i |\widetilde{\psi_i}\rangle \langle\widetilde{\psi_i}| = \sum_{ijk} u_{ij} u_{ik}^{*}|\widetilde{\phi_j}\rangle \langle\widetilde{\phi_k}|$$
In equation 2.168 adjoint of the tilded psi has now the element in the unitary matrix u being ik conjugated($u_{ik}^*$). Now I understand that the column index after the adjoint will not be the same due to the transpose(hence k instead of j), what I don't understand is why the row index (i) is unchanged. I know it's probably something simple that I am missing, but I would appreciate your help.