∀ {α : Type u_1} [inst : UniformSpace α] {β : Type u_2} [inst_1 : UniformSpace β] {f : α → β}
  {g : UniformSpace.Completion α → UniformSpace.Completion β},
  UniformContinuous g → (∀ (a : α), ↑(f a) = g ↑a) → UniformSpace.Completion.map f = g