Uniqueness of completion for metrically generated constructs

In this paper, for metrically generated constructs
$\X$ in the sense of \cite{mgt} we study
completion as a $\U$-reflector $R$ on the
subconstruct $\X_0$ of all $T_0$-objects, for $\U$
some class of embeddings. Roughly speaking we
deal with constructs $\X$ that are generated by
the subclass of their metrizable objects and for
various types of completion functors $R$
available in that context, we obtain an internal
descriptions of the largest class $\U$ for which
completion is unique.
We apply our results to some well known
situations. Completion of uniform spaces, of
proximity spaces or of non-Archimedean uniform
spaces is unique with respect to the class of all
epimorphic embeddings, and this class is the
largest one. However the largest class of
morphisms for which Dieudonn\'e completion of
completely regular spaces or of zero dimensional
spaces is unique, is strictly smaller than the
class of all epimorphic embeddings. The same is
true for completion in quantitative theories like
uniform approach spaces for which the largest
$\U$ coincides with the class of all embeddings
that are dense with respect to the metric
coreflection. Our results on completion for
metrically generated constructs explain these
differences.}