Let $\pi$ and $\sigma$ be representations of a $C^*$-algebra $\mathcal{A}$. They are weak approximately equivalent ($\pi\mathbin{\sim_{\rm wa}}\sigma$) if there are sequences of unitary operators $\{U_n\}$ and $\{V_n\}$ such that \begin{equation} \sigma(A)=\operatorname{WOT-lim} U_n\pi(A) U_n^*, \end{equation} \begin{equation} \pi(A)=\operatorname{WOT-lim} V_n\sigma(A) V_n^* \end{equation} for all $A\in\mathcal{A}$.
Many books point out that both directions are needed to obtain an equivalence relation but no clue is given why. Since for approximate equivalence ($\mathbin{\sim_{\rm a}}$), only one direction is needed, I wonder why for $\mathbin{\sim_{\rm wa}}$ we need two.
Thanks!