Consider the full subcategory $\mathcal C$ in the big etale site of a field $k$ consisting of smooth schemes, namely the category of smooth varieties over a field $k$ with etale maps as morphisms.

I am intersted in morphisms between constant functor $\mathcal{C}\to Vect_k$ which maps all objects to $k$ and all morphisms to identity and the functor of algebraic de Rham cohomology $X\mapsto H^i_{dR}(X)$.

Explicitly, this means fixing a class $c_X\in H^i_{dR}(X)$ for every $X$ such that $f^*c_Y=c_X$ for any etale $f:X\to Y$. For even $i$ Chern classes of tangent bundle give an example(since tangent bundle is preserved under etale morphisms).

Are there any nonzero functorial classes for odd $i$? Is it true that any functorial class in even cohomology is a polynomial in Chern classes?