The topic of Bousfield localizations has a lot of literature which has on most of the occasion some tameness assumption on the presentability of the model category. Recently I have been trying to **avoid any tameness** assumption but I can't avoid a couple of results about localizations. 


Thus, the content of this question is: (if true) **are the following two statements already in the literature**?



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> **Hope 1:** Let $\mathsf{M}$ and $\mathsf{N}$ be model categories and $\mathsf{L}: \mathsf{M} \leftrightarrows \mathsf{N}: \mathsf{R}$ be Quillen adjunction which specifies $\mathsf{N}$ as a reflective subcategory of $\mathsf{M}$, then $\mathsf{N}$ is Quillen equivalent to a Bousfield localization of $\mathsf{M}$, where the new weak equivalences $\bar{\mathcal{W}}$ are $\mathcal{W}\_{\mathsf{M}} \cup \mathsf{L}^{-1}(\mathcal{W}\_{\mathsf{N}})$.
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This should be seen as a technical improvement of **Rem. 3.8** on the nlab [page about idempotent monads](https://ncatlab.org/nlab/show/idempotent+monad#FactorizationOfIdempotentCore).



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> **Hope 2:** Let $\mathsf{M}$ and $\mathsf{T}$ be an idempotent monad over $\mathsf{M}$, then there exists a model structure on $\mathsf{Alg}(\mathsf{T})$ and a Quillen adjunction $\mathsf{L}: \mathsf{M} \leftrightarrows \mathsf{Alg}(\mathsf{T}): \mathsf{R}$ such that $\mathsf{Alg}(\mathsf{T})$ is Quillen equivalent to a Bousfield localization of $\mathsf{M}$.
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This would be a variant of **Prop. 3.8** in the [lecture notes](https://ncatlab.org/nlab/show/Introduction+to+Stable+homotopy+theory+--+1-1#ClassesOfMorphismsInBousfieldLocalizationAtQuillenIdempotentMonad) by Urs Schreiber where the model structure is not assumed to be right proper and the notion of idempotent monad is strictified. In **Def. 3.3**, while I really like that $\mathsf{T}$ does not have to be a monad, I find quite unnatural the condition (3) and the assumption that the structure is right proper in **Prop. 3.8**. 




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It is fine to assume that $\mathsf{M}$ is right proper if needed, but I cannot expect the adjunction to *respect* it.



Transfer model structures, reflective subcategories and Bousfield localizations
So far, the best results that I managed to find in the direction of my questions appear both in some paper of Boris Chorny and his collaborators.




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**1st. Theorem, A non-functorial Bousfield-Friedlander localization**,


This result appears as **A.8** in the paper below, it is a bit too long to be cast in this answer, in a nutshell it provides a version of the **Hope 2** when $\mathsf{T}$ is not even a functor.



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> **Duality and small functors**, *B. Chorny and G. Biedermann*. Algebr. Geom. Topol. 15 (2015) 2609-2657
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**2nd. Theorem, A generalization of Smith's theorem,**



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> **Thm 3.14** (C.R.) Let $\mathsf{M}$ a nice, proper, simplicial class-combinatorial model category and $\mathsf{T} : \mathsf{M} \to \mathsf{M}$ a strongly class-accessible homotopy idempotent functor preserving weak equivalences. Suppose additionally, that pullbacks of $\mathsf{T}$-equivalences along $\mathsf{T}$-fibrations are $\mathsf{T}$-equivalences. Then the $\mathsf{T}$-localization exists and is class-combinatorial.
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The main feature of this theorem is that **the localization is class-combinatorial**.


This result appears in,



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> **Class combinatorial model categories**, *B. Chorny and J. Rosický*, Homology Homotopy Appl. 14 (2012), no. 1, 263--280
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I recommend the reader not to be treated by the use of the world *accessible* and *locally presentable*. **Class-accessibility** is a very strong weakening of the notion of accessibility. On one hand, it **escapes the world of categories with a dense (small) generator**, on the other, it still allows to **build** on the technical power of the **small object argument**.


Another paper that should be mentioned is 



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> **Locally class-presentable and class-accessible categories**, *B. Chorny and J. Rosický*, J. Pure Appl. Alg. 216 (2012), 2113-2125.
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where the general theory of class-accessibility and class-local presentability is discussed.


