I am trying to prove that a functor $\mathcal{F}:(Sch/S)^{op}\rightarrow (Sets)$ is representable by an $S$-scheme $F$. My intuition says that it is indeed representable. I have been reading Fulton's "Hurwitz schemes and irreducibility of moduli of algebraic curves", where he uses Corollary 2 of "Representation of unramified functors" by Murre to show that a functor $\mathcal{H}:(Sch/S)^{op}\rightarrow (Sets)$ is representable by a scheme $H$ which is étale over $P$.
However, I know that if $F$ exist it cannot be étale over $S$ ($S$ and the hypothetical $F$ would be the same dimension, but the fibers of the structure morphism would vary in dimension). My question is:
Is there some general criteria to show that such an $\mathcal{F}$ is representable by a scheme $F$ which is not necessarily étale over $S$? I found some criteria for $\mathcal{F}$ to be representable by an algebraic space, but I am looking for this stronger representability result.
Thanks