because [@DI] is only a slight improvement of Part I of [@Dingyu] while having tikZ diagrams and being more accessible through arXiv, I will henceforth only refer to results in [@DI] (rather than mentioning both [@Dingyu] and [@DI] all the time) despite the fact that [@Dingyu] appeared earlier (and now made available on my webpage). [@DII] and [@DIII] will be substantially expanded based on Part II of [@Dingyu] to include more related results, so that the forgetful and globalization functors will enjoy more properties.

First, let me say a word about my approach of Kuranishi structures in [@DI]. The starting point of my theory of Kuranishi structures is a modified version of FOOO’s Kuranishi structures presented in [@FOOO], so that it has better properties, e.g. easy to obtain a good coordinate system from it (almost like obtaining a finite cover for a compact topological space), and then just sufficiently precompactly shrink a good coordinate system (in a strong shrinking sequence), one can get a Hausdorff one to further construct things (level-1 structure, perturbation, fiber product, etc), and this easy process to achieve a Hausdorff good coordinate system is crucial for the **functoriality** aspect. The modification is done by imposing two 