The injective hull of ultra-quasi-metric versus q-hyperconvex hull of quasi-metric space

For any partially ordered set equipped with its natural T0T0-quasi-metric (T0T0-ultra-quasi-metric), we study the connection between the ultra-quasi-metrically injective hull and the q  -hyperconvex hull. We also observe that for a partially ordered set, its Dedekind–MacNeille completion coincides exactly with the ultra-quasi-metric injective hull of its natural T0T0-ultra-quasi metric.