The maintenance of homotopy invariance for presheaves with Witt-transfers under Nisnevich sheafication

In the present paper, we introduce a category $Wor$, whose objects are affine smooth varieties over a field $k$ and morphisms are a certain variant of finite correspondences. A presheaf of abelian groups with Witt-transfers is by definition a presheaf of abelian groups on the category $Wor$. We prove the homotopy invariance of the Nisnevich sheaf associated with an arbitrary homotopy invariant presheaf with Witt-transfers. To construct a category of Witt-motives one should prove the homotopy invariance of Nisnevich cohomology of an arbitrary homotopy invariant Nisnevich sheaf with Witt-transfers.