Weil restriction and the motivic cycle class map

We construct the Weil restriction map for l-adic cohomology and, more generally, for mixed Weil cohomology theories. We study its compatibility with the motivic cycle class map and show that these constructions admit a natural interpretation in the triangulated categories of motives. Using Grothendieck's six-functor formalism, we prove that the Weil restriction map arises intrinsically from the functorial structures of these categories. This provides a conceptual framework for understanding the interaction between Weil restriction, motivic cohomology, and realization functors.