Equivalence of coherent theories

In a series of papers Yu. Lisica and S. Mardešić developed a coherent theory of inverse systems X = (Xa, pa0,a1, A), where for a0 < a1 < a2, pa0a1pa1a2 = pa0a2, and the corresponding strong shape theory. A coherent map between two systems ƒ: X → Y is defined by homotopies of all orders. Category CPHTop has as objects inverse systems X and as morphisms homotopy classes of coherent maps. general type of inverse systems are the systems χ = (Xa, pa0…an, A) where the equation above is replaced by existence of a homotopy pa0a1a2: I × Xa2 → Xa0 connecting maps pa0a1pa1a2 and Pa0a2, but also in this system there exist homotopies pa0…an: In − 1 × Xan → Xa0 of arbitrary order which on the boundary ∂In − 1 × Xan are defined by homotopies of lower order. The coherent category of these systems is denoted by Coh. e systems X can be considered as objects of Coh and actually CPHTop is a subcategory of Coh. In this paper we positively solve the problem: Is CPHTop a full subcategory of Coh?