$\newcommand{\Z}{\mathbb{Z}}$ $\newcommand{\F}{\mathbb{F}_1}$ $\newcommand{\spec}{\operatorname{Spec}}$

If I understand correctly, in Borger's paper about the field with one element, the category of "affine schemes over $\spec\F$" is just the opposite of the category of $\Lambda$-rings. The base change from $\spec\F$ to $\spec\Z$ is forgetting the $\Lambda$-structure. i.e. we can view a $\Lambda$-structure on a commutative ring as a descent data from $\spec \Z$ to $\spec\F$ (there is ageneralization to the non-affine case, but let's keep it simple for now).

The base change has adjoints from both sides. The one that comes from the composition $$\operatorname{Spec}R \to \operatorname{Spec}\mathbb{Z} \to \operatorname{Spec}\mathbb{F}_1$$

Takes a ring $R$ to the $\Lambda$-ring of (big) Witt vectors over $R$ denoted $W(R)$.

Now, it seems to me that a desirable property of this setup would be that an (affine) scheme over $\spec\Z$ would be "the same as" an (affine) scheme over $\spec\F$ with a morphism to $\spec\Z$ over $\spec\F$. So for a $\Lambda$-ring $R$, a map of $\Lambda$-rings $W(\Z)\to R$ should be induced by a ring $R_0$ by applying $W$ to the structure morphism $\Z \to R_0$. Moreover, there should be a natural bijection between the two types of data (up to isomorphisms in the obvious way). It seems that the elements of $R_0$ should be something like elements of $R$ of rank $\le 1$ (i.e. those for which $\lambda^n$ vanish for n $> 1$) and the sum should be something like the "rank one approximation of the sum in $R$).

Question: Is this indeed true?

Perhaps I am wrong in assuming that this is desirable, so an explanation of that would be great too. By the way, I am not an expert on $\Lambda$-rings and I ask this purely out of curiosity.