  By introducing the concept of quantaloidal completions for an order-enriched
category, relationships between the category of quantaloids and the category of
order-enriched categories are studied. It is proved that quantaloidal
completions for an order-enriched category can be fully characterized as
compatible quotients of the power-set completion. As applications, we show that
a special type of injective hull of an order-enriched category is the MacNeille
completion; the free quantaloid over an order-enriched category is the Down-set
completion.
