Sorry in advance if my question doesn't have the level of this community.

I am studying this paper of Bondal and Van Den Bergh and in particular section 2. Generators and resolutions in triangulated categories.

As long as I was figuring out the definitions of "classically generated", "generated" and "compactly generated", I came up with the Theorem 2.1.2 of this paper which says that in a compactly generated category $\mathcal{C}$, a set of objects $\mathcal{E}\subset \mathcal{C}^{c}$ classically generates $\mathcal{C}^{c}$ if and only if it generates $\mathcal{C}$. For the proof I started to read the reffering paper by Neeman and successfully finished and understood it. But there are some problems of connecting the information of the 2 papers.

- First of all, while theorem 2.1.2 uses the notion of "generates" that Bondal and Van Den Bergh define, in Neeman's paper there is nowhere such a notion and
**maybe**instead of this, he uses the notion of the smallest localising triangulated subcategory that contains a set of objects. I think that there is the following connection which I unfortunately can't prove $$\mathcal{R}\ \text {is the smallest localising triangulated subcategory that contains a set}\ R\subset \mathcal{C}^{c}\ \Leftrightarrow $$$$R^{\perp}=0 $$ where $R^{\perp}$ is defined as in Bondal's paper as a subcategory of $\mathcal{R}$ and $\mathcal{R}$ is the smallest localising triangulated subcategory that contains a set $R\subset \mathcal{C}^{c}$ as in Neeman's paper. I strongly think that this is true under the hypothesis that $\mathcal{R}\subset \mathcal{C}$ where $\mathcal{C}$ is compactly generated. - If the 1. is true then I construct the proof of Theorem 2.1.2 in the following way: by lemma 2.2 in Neeman's Paper(in the main proof of theorem 2.1) we clearly have the implication $\mathcal{E}$ generates $\mathcal{C}$ $\implies$ $\mathcal{E}\subset \mathcal{C}^{c}$ classically generates $\mathcal{C}^{c}$. For the other direction we note that if the smallest localising triangulated subcategory that contains $\mathcal{C}^{c}$ is the whole $C$, then $\mathcal{C}$ consists of coproducts of objects in $\mathcal{C}^{c}$. Then it is easy, since $Hom(M,\bigoplus E_{i})\cong \bigoplus Hom(M,E_{i})$.

As it seems there is a problem of combining these two papers and any help will be accepted.

**Edit**: To be honest I hadn't found this post that has an answer from Leonid Positselski, so I reform my question: as the proof is for well-generated categories (a generalised notion of compact objects) and uses the Brown representability theorem for triangulated categories I was wondering if there is a simpler proof for compact objects without using it. Maybe the first proof before the introduction of well-generatedness by Krause and others.