For a given ∼R, let R be its set of equivalence classes. This induces maps x:X→R and y:Y→R. The isomorphism f:X→Y you discuss has the property y⋅f=x. Maps like that are the morphisms in the slice category over R, and these isomorphisms are the isomorphisms in that category. So what happened is that you’ve given X and Y the structure of a bundle, and the isomorphisms respect that structure.
For a given ∼R, let R be its set of equivalence classes. This induces maps x:X→R and y:Y→R. The isomorphism f:X→Y you discuss has the property y⋅f=x. Maps like that are the morphisms in the slice category over R, and these isomorphisms are the isomorphisms in that category. So what happened is that you’ve given X and Y the structure of a bundle, and the isomorphisms respect that structure.