conditions to FOOO’s Kuranishi structures and they are called the maximality condition and the topological matching condition; and they are non-destructive to the original definition, as my goal was to unify HWZ’s theory of polyfolds/polyfold Fredholm structures and FOOO’s theory of Kuranishi structures after all, and this approach pays off in \[FOOOISDIR\] which essentially says that the theory applies to FOOO’s Kuranishi structures as well. A key point of the theory in [@DI] is that one should **view Kuranishi structures on the level of R-equivalences** (admitting a common refinement), e.g. perturbation theory factorizes through it, and the importance of this notion is also illustrated by the fact that following a loop of arrows (in the arrow direction) back to starting virtual structure in the unification diagram \[UNIFDIAG\], the effect is identity up to R-equivalence of that said structure (or effect is the identity morphism between R-equivalence classes). **The global nature of R-equivalence has the advantage that it has no structural damage to the virtual structure, and if one would like, he can also work on the level of Kuranishi structures, do things up to R-equivalence.** R-equivalence is more appealing and easier for functoriality if we have a common refinement (more 