Mappings and isometries of compact metric spaces

In the paper it is proved that there exists a continuous surjective mapping F:X→Y, where X and Y are separable complete metric spaces of transfinite dimension ind less than or equal to αd∈ω+∪{∞} and αr∈ω+∪{∞}, respectively, such that, for each continuous surjective mapping f:X→Y, where X and Y are compact metric spaces of transfinite dimension ind less than or equal to αd and αr, respectively, there are isometries i:X→X and j:Y→Y satisfying the relation F∘i=j∘f. This result remains true if we suppose that, for each mapping f:X→Y, Y coincides with a fixed separable complete metric space Y and that the corresponding mapping j:Y→Y is the identity.