The category of elements and section preorders

The category of elements of a copresheaf $F$ denoted $el(F)$ provides an algebraic context to the section preordering. In particular, we have that the section preordering on $F$ is just the object preordering of $el(F)$.

Proposition. let $F : C \to Sets$ be a copresheaf, then the object preordering of $el(F)$ is the section preordering of $F$.

This is useful because now we know that the section preordering of $F$ can be constructed from the object preorder of its category of elements. The subobject lattice of $F$ is then basically the Heyting algebra of the Alexandrov topology of open subsets of the section preorder.

See also:
Subobject lattices of presheaves