It is known that given a uniform space $(X, \mathcal{U})$ and a function $\pi$ from $X$ onto an arbitrary set $Y$, it is possible to define a uniformity on $Y$, in the following way: \begin{equation*} \mathcal{U}_\pi= \{B: B\subseteq Y\times Y: \pi^{-1}(B)\in\mathcal{U}\}. \end{equation*} The uniformity $\mathcal{U}_{\pi}$ is called the quotient uniform with respect to $\pi$ and also we can say that \begin{equation*} \mathcal{U}_{\pi}=\{\pi\times \pi(U): U\in\mathcal{U}\}. \end{equation*}
Q. Is it true that for every $U\in\mathcal{U}$, there is $V\in\mathcal{U}_\pi$ such that if $(\pi(x),\pi(y))\in V$, then $(x, z)\in U$ for some $z\in\pi(y)$.