ag.algebraic geometry - Endofunctors of CRing which give schemes when composed with schemes?

Here is a nontrivial example I like. Let $W:mathrm{Rings}tomathrm{Rings}$ denote the Witt vector functor of some fixed finite length. (You can consider the $p$-typical Witt vectors, for some prime $p$, but everything works with the other standard flavors.) Then the functor $W_*(-)=-circ W$ is an endofunctor of the category of functors $mathrm{Rings}tomathrm{Sets}$, and it takes schemes to schemes. The scheme $W_*(X)=Xcirc W$ is the so-called arithmetic jet space of $X$, extensively studied by Buium in the case of $p$-adic formal schemes. The fiber over $p$ is the Greenberg transform.

There is a standard method for proving $W_*$ takes schemes to schemes (or rather for proving almost that), though it probably doesn't work for every functor $F:mathrm{Rings}tomathrm{Rings}$ such that $F_*$ takes schemes to schemes. First you show that the category of sheaves of sets on the category of affine schemes w.r.t. the etale topology is stable under $W_*$. This is true since $W$ takes etale covers of rings to the same and also takes cocartesian squares of etale rings maps to the same. (These properties of $W$ are not obvious.) Then you show $W_*$ takes sheaf epimorphisms to the same, and etale maps of sheaves to the same. (These properties are much easier.) Therefore any etale equivalence relation on an affine scheme is sent to an etale equivalence relation on an affine scheme, and the quotient of the first is sent to the quotient of the second. Therefore the category of quasi-compact quasi-separated algebraic spaces is stable under $W_*$. I have no doubt you could find reasonable abstract properties on the endofunctor $W$ of Rings that allow this argument to go through.

Showing $W_*$ takes schemes to schemes is a bit subtler. You have to deal with quasi-compactness issues (as a right adjoint, $W_*$ doesn't necessarily behave well w.r.t. disjoint unions) and also the fact that it's harder to tell whether a functor is represented by a scheme than by an algebraic space. But as I said, in the Witt vector example above, it is true.

Presumably the same argument works, and is much easier, for the functor $F$ defined by $F(R)=R[t]/(t^{n+1})$. Then $F_*(X)$ should be the usual jet space functor of length $n$. The case $n=1$ should give the total space of the tangent bundle, at least when $X$ is smooth.

Edit: I'm reminded below that this example is just a particular case of the representability of the Weil restriction of scalars for a finite flat map $Ato B$. There you consider the endofunctor of the category of $A$-algebras given by $F(R)=Botimes_A R$. In particular, it's reasonable to view $W_*$ as a generalized Weil resitrction of scalars.