Here on the page 2, I cannot find the definitions of $[\operatorname{BRM}]$ and $[\operatorname{BRM}_\lambda]$.
Are they in the present paper ? I've also looked at the references at the end but couldn't find nothing helpful.
I understand from the paper some equivalents of these conditions but not the definitions.
EDIT:
As suggested by John Palmiery I've looked up [11] but still haven't caught the right meaning:
they say the category $\cal T$ satisfies [BRM] if every n.t. $\mathbf{y} X\to \mathbf{y}X'$ is induced by a map $X\to X'$. My problem is twofold, what is $\mathbf y$ and in what sense induced is meant here.
My guess is that $\mathbf{y}$ stands for "Yoneda".
As given in [11] (Christensen-Keller-Neeman, "Failure of Brown Representability in Derived Categories", https://arxiv.org/abs/math/0001056), for an object $X$ in the category $\mathcal{T}$, $\mathbf{y} X$ is the functor $\mathcal{T}(-, X)$ but restricted to the category of compact objects of $\mathcal{T}$. Given a map $X \to X'$, there is an induced natural transformation $\mathbf{y}X \to \mathbf{y}X'$; the requirement for [BRM] is that every natural transformation arises in this way.
Edit: given $f: X \to X'$, we get a natural transformation $\mathbf{y}X \to \mathbf{y} X'$ as follows: $g \in \mathcal{T}(A, X)$ gets sent to $f \circ g \in \mathcal{T}(A, X')$.