If $X$ is a set equipped with a collection $(d_i)_{i\in I}$ of pseudometrics, then the corresponding uniform structure (collection of entourages) $\Phi$ is defined by declaring that $U\in\Phi$ if and only if there exists $n\in\mathbb N$, $i_1,\ldots,i_n\in I$, and $\epsilon_1,\ldots,\epsilon_n>0$ such that $B(i_1,\epsilon_1)\cap\cdots\cap B(i_n,\epsilon_n)\subseteq U$, where $B(i,\epsilon)=\{(x,y)\in X\times X:d_i(x,y)<\epsilon\}$.
How is the converse defined? That is, given a uniform structure, what is the corresponding collection of pseudometrics?
I think "uniform structure $\to$ collection of pseudometrics $\to$ uniform structure" should yield a space uniformly isomorphic to the original space. Also, "collection of pseudometrics $\to$ uniform structure $\to$ collection of pseudometrics" should do the same.