Let $U : \mathbf{CRing} \to \mathbf{Set}$ be the forgetful functor. For any functor $F : \mathbf{CRing} \to \mathbf{Set}$ consider the class of natural transformations $$\mathcal{O}(F) := \mathrm{Hom}(F,U).$$ It is clear that $\mathcal{O}(F)$ carries the structure of a commutative "ring" (it doesn't have to be a set). When $F$ is representable, say $F \cong \mathrm{Hom}(S,-)$, then $ \mathcal{O}(F) \cong S$ by the Yoneda Lemma.

More generally, in functorial algebraic geometry, we can consider schemes as (special) functors $F : \mathbf{CRing} \to \mathbf{Set}$, and $\mathcal{O}(F) = \mathrm{Hom}(F,\mathbb{A}^1)$ is the ring of global sections of $F$. In particular, this is a set. It follows that this also holds for algebraic spaces.

Question. Is there a classification of those functors $F : \mathbf{CRing} \to \mathbf{Set}$ for which the class $\mathcal{O}(F)$ is actually a set?

If this is not solvable: Is there a classification of the continuous functors $F$ with the property? At the nlab page on total categories there is an example of a continuous functor $F$ which is not representable. If this is also not possible, I am looking for sufficient conditions. Although I wrote about classes, I am also very up for using universes instead. But I hope that the answer to my question does not depend on some subtle set theory axioms.