What's the relation between the pointed motivic homotopy category $\mathcal{H}_*(k)$ and the derived category of motives $\mathbf{DM}^-_{eff}(k)$ besides the representability of motivic cohomology in the homotopy category?
I think there is a functor $\mathcal{H}(k) \to \mathbf{DM}^-_{eff}(k)$. How far is it from being full and faithful?
What's the relation between the pointed motivic homotopy category $\mathcal{H}_*(k)$ and the derived category of motives $\mathbf{DM}^-_{eff}(k)$ besides the representability of motivic cohomology in the homotopy category?