Generalized Brown representability in homotopy categories

We show that the homotopy category of a combinatorial stable model category $\text{ck}$ is well generated. It means that each object $K$ of $\text{Ho}(\text{ck})$ is an iterated weak colimit of $\lambda$-compact objects for some cardinal $\lambda$. A natural question is whether each $K$ is a weak colimit of $\lambda$-compact objects. We show that this is related to (generalized) Brown representability of $\text{Ho}(\mathcal{K})$.