I thought the following facts about $\lambda$-Rings were all true, but it seems like they yield a contradiction, so something is awry. Let $U$ be the forgetful functor $\lambda-Ring\rightarrow Ring$, and $W$ is functor of Witt vectors. We take the arithmetic viewpoint that $\lambda$ rings are rings with families of commuting frobenius lifts.
1) We have an adjunction $hom_{Ring}(U(A),B)\cong hom_{\lambda-Ring}(A,W(B))$
2) Any map of torsion free $\lambda$ rings commutes with the frobenius lifts (that define the $\lambda$ structure).
3) The normalised chebyshev polynomials on $\mathbb{Z}[x]$ give a $\lambda$-structure to $\mathbb{Z}[x]$. eg, $\Psi^2(x)=x^2-2,\Psi^3(x)=x^3-3x$
4) There are no nontrivial endomorphisms of the $2$-adic integers $\mathbb{Z}_2$.
The problem arises when we let $A=\mathbb{Z}[x]$ with this $\lambda$ structure, and $B=\mathbb{F}_2$.
The left hom-set has two elements, as we can map $x$ to $0,1$ in $\mathbb{F}_2$.
However, by 4), the Adams operations on $\mathbb{Z}_2$ are trivial, so by 2) and 3), we see that the image of $x$ must be a root of $x^2-x-2$ for a $\lambda$ morphism in the right hand hom-set. In order to have two morphisms, sending $x$ to the roots of this polynomial, $-1$ and $2$ must both yield $\lambda$ morphisms $\mathbb{Z}[x]\rightarrow \mathbb{Z}_2$. However, for $x\mapsto -1$, we see that the Adams operation $\Psi^3$ does not respect this map, if $\phi$ is the map, then: $$\phi(\Psi^3(x))=\phi(x^3-3x)=(-1)^3+3=2$$ While we have: $$\Psi^3(\phi(x))=\Psi^3(-1)=-1$$
So something here is false, but I can't see what it is.
The mistake was in $W(\mathbb{F}_2)\cong \mathbb{Z}_2$, this holds for $2$-typical Witt vectors ,but the big Witt vectors on $\mathbb{F}_2$ is much bigger than this, and allows for nontrivial frobenius lifts.