Summary: A bornology \(\mathcal{B}\) on a set \(X\) is called minmax, if the smallest and 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 \(X\). In addition, 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.