The completion of a metric space is unique up to metric monomorphism (usually called isometry).
It is also the "obvious" way to make all Cauchy sequences convergent.
Structures which are unique up to structure-preserving isomorphism and which define some type of canonical or best way of modifying some other structure are usually characterizable in terms of universal properties.
Question: In what category (i.e. describe both the objects and morphisms) can the completion of a metric space be considered a universal object? (and please give an explicit formulation of the universal property for which the completion is the universal object.)
If no such category exists, explain why.
A similar question has been asked here before Completion of a metric space in categorical terms; however I am not satisfied with the accepted answer, since it just parrots what Wikipedia says and gives no justification (and the answer given on Wikipedia has no source attribution, nor is any proof or justification attributed to the answer); the only other answer only addresses the issue in terms of enriched categories, which seems like an unnecessarily complicated way to characterize the issue.
A similar question was also asked on MathOverflow, but the answers given to that question were also inadequate, since they lack adequate justification or explanation of their claims, and don't provide any comprehensive resources (something like this is mentioned briefly as an example in MacLane's Categories for the Working Mathematician, but it really isn't adequately well-posed).
All of the results I have found on the internet (excluding the answers on the two aforementioned StackExchange network, which more or less parrot Wikipedia's unjustified and citation-less claim) all give seemingly contradictory characterizations of metric space completion in terms of universal properties. For example, Theorem 16 of this page characterizes it in terms of isometries; this page uses Cauchy continuous functions; again Wikipedia claims there is one in terms of uniformly continuous functions and uniform structure without citation or proof.
Perhaps it is the case that all three of these characterizations are equivalent -- if so, I would I would greatly appreciate an explanation of how this is the case. Also Lipschitz or Holder continuous functions also seem like candidates for the morphisms in the metric space category for which metric space completions are a universal object, in addition to the three types of maps mentioned above.