Summary: We consider compatible monoid structures on a \(\mathsf{DQ}\)-category, where \(\mathsf{Q}\) is a commutative and divisible quantale, and \(\mathsf{DQ}\) is the quantaloid of diagonals of \(\mathsf{Q}\). Such structures, called \(\mathsf{DQ}\)-monoids, may be treated as monoids equipped with a preorder valued in \(\mathsf{Q}\), whose elements are not supposed to exist globally. The globalization functors map \(\mathsf{DQ}\)-monoids to global ones, i.e., monoids on a \(\mathsf{Q}\)-category, called \(\mathsf{Q}\)-monoids. Conversely, \(\mathsf{Q}\)-monoids over \(\mathsf{Q}\) give rise to \(\mathsf{DQ}\)-monoids via the localization functors. The interactions between the globalization functors and the localization functors are investigated. In particular, a necessary and sufficient condition on \(\mathsf{Q}\) is provided such that the localizations are reversible by the globalizations.