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**?

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 $\mathcal{W}_{\mathsf{M}} \cup \mathsf{L}^{-1}(\mathcal{W}_{\mathsf{N}})$.

This should be seen as a technical improvement of **Rem. 3.8** on the nlab page about idempotent monads.

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 $\mathsf{M}$.

This would be a variant of **Prop. 3.8** in the lecture notes 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**.

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