Effective Grothendieck-Witt motives of smooth varieties

The category of effective Grothendieck-Witt-motives $\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ (and Witt-motives $\mathbf{DM}^W_{\mathrm{eff},-}(k)$) by Voevodsky-Suslin method starting with some category of GW-correspondences (and Witt-correspondences) over a perfect field $k$, $char\,k\neq 2$, is defined. The functor $M^{GW}_{eff}\colon Sm_k\to \mathbf{DM}^{GW}_{\mathrm{eff},-}(k)$ of Grothendieck-Witt-motives of smooth varieties is computed and it is proved that for any smooth scheme $X$ and homotopy invariant sheave with GW-transfers $F$ $$ Hom_{\mathbf{DM}^{GW}_{\mathrm{eff},-}(k)}(M^{GW}_{eff}(X), F[i]) \simeq H^i_{nis}(X, F) $$ naturally in $X$ and $F$.