  Properties of categories enriched over the category of metric spaces are
investigated and applied to a study of constructions known from that category
and the category of Banach spaces. For every class of morphisms satisfying a
mild smallness condition we prove the corresponding approximate-injectivity
class is weakly reflective, and we study the properties of the reflection
morphisms. As an application we present a new categorical proof of the
essential uniqueness of the Gurarii space.
