The monomorphisms in Met are the injective metric maps.
The isomorphisms are the isometries, i.e. metric maps which are injective, surjective, and distance-preserving.
Injective metric spaces were introduced and studied first by Aronszajn & Panitchpakdi (1956), prior to the study of Met as a category; they may also be defined intrinsically in terms of a Helly property of their metric balls, and because of this alternative definition Aronszajn and Panitchpakdi named these spaces hyperconvex spaces.
The forgetful functor Met → Set assigns to each metric space the underlying set of its points, and assigns to each metric map the underlying set-theoretic function.
This functor is faithful, and therefore Met is a concrete category.