The monomorphisms in Met are the injective metric maps, the epimorphisms are the dense image metric maps (for instance, the inclusion: , which is clearly mono, so Met is not a balanced category), and the isomorphisms are the isometries.
We have a "forgetful" functor Met → Set which assigns to each metric space the underlying set, and to each metric map the underlying function. This functor is faithful, and therefore Met is a concrete category.
The injective objects in Met are called injective metric spaces. They were introduced and studied first by Aronszajn and Panitchpakdi (1956), who named them hyperconvex spaces. Any metric space has a smallest injective metric space into which it can be isometrically embedded, called its metric envelope or tight span. It is related to the notion of the universal (or maximal) metric space aimed at its subspace.