Inverse limits of uniform covering maps

In [N. Brodskiy, J. Dydak, B. LaBuz, A. Mitra, Rips complexes and covers in the uniform category, preprint] the authors define, following James, covering maps of uniform spaces and introduce the concept of generalized uniform covering maps. Conditions for the existence of universal uniform covering maps and generalized uniform covering maps are given. This paper notes that the universal generalized uniform covering map is uniformly equivalent to the inverse limit of uniform covering maps and provides a characterization of generalized uniform covering maps that are uniformly equivalent to the inverse limit of uniform covering maps.