  A bornology $\mathcal{B}$ on a set $X$ is called minmax if the smallest and
the largest coarse structures on $X$ compatible with $\mathcal{B}$ coincide.
  We prove that $\mathcal{B}$ is minmax if and only if the family $\mathcal
B^\sharp=\{p\in\beta X:\{X\setminus B:B\in\mathcal B\}\subset p\}$ consists of
ultrafilters which are pairwise non-isomorphic via $\mathcal B$-preserving
bijections of $X$. Also we construct a minmax bornology $\mathcal B$ on
$\omega$ such that the set $\mathcal B^\sharp$ is infinite. We deduce this
result from the existence of a closed infinite subset in $\beta\omega$ that
consists of pairwise non-isomorphic ultrafilters.
