Let $C$ be the category of commutative rings.
Is there a functor $F :C \to C$ such that $F(F(R)) \cong R[X]$ for every commutative ring $R$ ?
(Here, we may assume those isomorphisms to be natural in $R$, if needed).
I tried to see what $F(\mathbb Z)$ or $F(k)$ (for a field $k$) should be, but I cannot come up with a contradiction to disprove the existence of $F$. On the other hand, I tried to build such an $F$, without success (e.g. try to consider some extension of $\mathbb Z[X_r : r \in R] / (X_{a+b} - X_a - X_b, X_{ab} - X_a X_b)$...).