Coherent homotopy and localization

The purpose of this paper is to give a direct and relatively simple proof of the fact that the coherent homotopy category CH(Top), introduced in 1983 by Yu.T. Lisitsa and S. Mardešić, is isomorphic to the category Ho(pro-Top), obtained by localizing the category pro-Top at level homotopy equivalences. This result implies that the strong shape category SSh(Top), introduced by these authors is isomorphic to the strong shape categories defined by F.W. Cathey and J. Segal and by B. Günther.