Let $X$ be a projective variety, and $E$ be a coherent sheaf on $X$. Grothendieck has proven that there is a scheme $\mathrm{Quot}\_X(E)$ parametrizing arbitrary quotient sheaves of $E$. It is probably a well-known question, but I found nothing in Google:


$$\text{Why not a scheme $\mathrm{Sub}\_X(E)$ parametrizing arbitrary subsheaves of $E$?}$$


Does such a scheme exist? If yes, is it purely a question of style, like Grothendieck's definition of the projectivization $\mathbb P(V)$ as the set of one-dimensional quotients of a vector space $V$? If not, is it because cokernels of sheaves require sheafification, while kernels coincide with kernels for pre-sheaves? 


Moreover, if one considers a reflexive sheaf $E$ (that is the canonical morphism $E \to E^{\*\*}$ is an isomorphism) and its reflexive subsheaves, then there is a Sub-scheme via dualizing and taking a Quot-scheme. Is there a less restrictive constraint?



Why there is a Quot-scheme, not a Sub-scheme?
For standard universal properties, you need the scheme to behave well under base change, which in these cases would mean tensor products. Tensor product is right exact, so a quotient remain a quotient, not left exact, so a sub may not remain a sub.


