I am reading Mac Lane's Categories for the Working Mathematician. He mentioned that the usual completion of metric space is universal for the evident forgetful functor (from complete metric spaces to metric spaces). (p.57)
I am not sure what is this forgetful functor 'forgetting'. I think this functor is still keeping the underlying set and also the metric. So does this functor only forget the concept of Cauchy sequence? If so, how to formulate this sentence rigorously? Thank's in advance!