  We show that for any quantale $\mathcal{Q}$, a $\mathcal{Q}$-category is
skeletal and complete if and only if it is injective with respect to fully
faithful $\mathcal{Q}$-functors. This is a special case of known theorems due
to Hofmann and Stubbe, but we provide a different proof, using the
characterisation of the MacNeille completion of a $\mathcal{Q}$-category as its
injective envelope. For Lawvere metric spaces, our results yield those of
Kemajou, K\"unzi and Otafudu. We point out that their notion of Isbell
convexity can be seen as a geometric formulation of categorical completeness
for Lawvere metric spaces.
