I recently found an answer to a similar question. Suppose:
$$(\forall V\in \mathcal U)(\exists W\in \mathcal U)(U\circ W\subseteq V\circ U)$$
By choice,for each $V \in \mathcal U$, some symmetric $D_V\in \mathcal U$ such that
$$D_V\circ D_V\subseteq  V$$

and there's some symmetric $W_V\in \mathcal U$ such that:

$$U\circ W_V \subseteq D_V\circ U$$
and
$$W_V\subseteq D_V$$

so
$$W_V\circ U\circ W_V \subseteq W_V\circ D_V\circ U\subseteq D_V\circ D_V\circ U\subseteq V\circ U$$

Therefore

$$\overline U=\bigcap_{W\in \mathcal U}W\circ U \circ W\subseteq \bigcap_{V\in \mathcal U}W_V\circ U \circ W_V\subseteq \bigcap_{V\in \mathcal U}V\circ U\subseteq U\circ U$$

Now see [this thread][1].


  [1]: http://math.stackexchange.com/questions/300376/is-overlined-subseteq-d-circ-d-in-a-uniform-space/350024