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][1], 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][2] 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][3] instead. But I hope that the answer to my question does not depend on some subtle set theory axioms.

  [1]: https://mathoverflow.net/questions/68936/source-on-functorial-algebraic-geometry
  [2]: https://ncatlab.org/nlab/show/total+category
  [3]: https://en.wikipedia.org/wiki/Tarski%E2%80%93Grothendieck_set_theory