version of a construction of this envisioned Polyfold Fredholm Homology (PFH) [@Yang], which is still not chain-level commutative yet due to infinite dimensionality. So with this PFH, a natural question would be how PFH and KH are related. This led to my effort in [@Dingyu] to functorially identify polyfold Fredholm structures and Kuranishi structures as geometric structures in a way intertwining respective perturbation theories (intertwining perturbation theories is not needed for PFH = KH, but is important in unifying structures in virtual theories devised for perturbations). This polyfold–Kuranishi correspondence [@Dingyu] has many nice properties (one reason for delay of my papers is that I want to have a good theory of structures at the output of functors and I want to get more properties for the functors); and I want to emphasize that the content is not trying to cast polyfold perturbation theory as a Kuranishi structure theory, instead I start from understanding an abstract Kuranishi structure on its own right and figure out a functorial way to convert between these two structures making both theories compatible and sometimes providing some new (even though so far just academic) understanding for each theory (e.g. two ways of perturbations in FOOO’s Kuranishi 