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The cancellation theorem for Grothendieck-Witt-correspondences andWitt-correspondences between smooth varieties over an infinite prefect field$k$, $char k \neq 2$, is proved, the isomorphism$$Hom_{\mathbf{DM}^\mathrm{GW}_\mathrm{eff}}(A^\bullet,B^\bullet) \simeqHom_{\mathbf{DM}^\mathrm{GW}_\mathrm{eff}}(A^\bullet(1),B^\bullet(1)),$$ for$A^\bullet,B^\bullet\in \mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)$ in thecategory of effective Grothendieck-Witt-motives constructed in\cite{AD_DMGWeff} is obtained (and similarly for Witt-motives).  This implies that the canonical functor $\Sigma_{\mathbb G_m^{\wedge1}}^\infty\colon \mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)\to\mathbf{DM}^\mathrm{GW}(k)$ is fully faithful, where$\mathbf{DM}^\mathrm{GW}(k)$ is the category of non-effective GW-motives(defined by stabilization of $\mathbf{DM}^\mathrm{GW}_\mathrm{eff}(k)$ along$\mathbb G_m^{\wedge 1}$) and yields the main property of motives of smoothvarieties in the category $\mathbf{DM}^\mathrm{GW}(k)$: $$Hom_{\mathbf{DM}^\mathrm{GW}(k)}(M^{GW}(X), \Sigma_{\mathbb G_m^{\wedge1}}^\infty\mathcal F[i]) \simeq H^i_{Nis}(X,\mathcal F) ,$$ for any smoothvariety $X$ and homotopy invariant sheave with GW-transfers $\mathcal F$ (andsimilarly for $\mathbf{DM}^\mathrm{W}(k)$).

Cancellation theorem for Grothendieck-Witt-correspondences and Witt-correspondences