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extracted a digest of the first theorem of Renaudin 06:
The 2-localization
$CombModCat\big[\{QuillenEquivalences\}^{-1}\big]$of the 2-category of combinatorial model categories at the Quillen equivalences exists. Up to equivalence of 2-categories, it has the same objects as $CombModCat$ and for any $\mathcal{C}, \mathcal{D} \in CombModCat$ its hom-category is the localization of categories
$CombModCat\big[\{QuillenEquivalences\}^{-1}\big](\mathcal{C}, \mathcal{D}) \;\simeq\; ModCat( \mathcal{C}^p, \mathcal{D}^p )\big[\{QuillenHomotpies\}^{-1}\big]$of the category of left Quillen functors and natural transformations between local presentations $\mathcal{C}^p$ and $\mathcal{D}^p$ at those natural transformation that on cofibrant objects have components that are weak equivalences (“Quillen homotopies”).
Added several references.
Added my own (draft of the) MR review
MR4112764 (zoranskoda) M. E. Descotte, E. J. Dubuc, M. Szyld , A localization of bicategories via homotopies, Theory and Applications of Categories 35, 2020, No. 23, 845–874, TAC
Also removed word “evident” from the idea section as there are several quite different nontrivial versions and some set theoretical issues when studied carefully.
Added link localization of an enriched category.
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