  Given a perfect field of exponential characteristic $e$ and a functor
$f:\mathcal A\to\mathcal B$ between symmetric monoidal strict $V$-categories of
correspondences satisfying the cancellation property such that the induced
morphisms of complexes of Nisnevich sheaves
  $$f_*:\mathbb Z_{\mathcal A}(q)[1/e]\to\mathbb Z_{\mathcal B}(q)[1/e],\quad
q\geq 0,$$ are quasi-isomorphisms, it is proved that for every $k$-smooth
algebraic variety $X$ the morphisms of twisted motives of $X$ with $\mathbb
Z[1/e]$-coefficients
  $$M_{\mathcal A}(X)(q)\otimes\mathbb Z[1/e]\to M_{\mathcal
B}(X)(q)\otimes\mathbb Z[1/e]$$ are quasi-isomorphisms. Furthermore, it is
shown that the induced functors between triangulated categories of motives
  $$DM_{\mathcal A}^{eff}(k)[1/e]\to DM_{\mathcal B}^{eff}(k)[1/e],\quad
DM_{\mathcal A}(k)[1/e]\to DM_{\mathcal B}(k)[1/e]$$ are equivalences. As an
application, the Cor-, $K_0^\oplus$-, $K_0$- and $\mathbb K_0$-motives of
smooth algebraic varieties with $\mathbb Z[1/e]$-coefficients are locally
quasi-isomorphic to each other. Moreover, their triangulated categories of
motives with $\mathbb Z[1/e]$-coefficients are shown to be equivalent. Another
application is given for the bivariant motivic spectral sequence.
