The presheaf of sieves yields the subobject classifier in the topos of presheaves. What about the presheaf of cosieves ? Does it have any interesting properties?
Let C be a small category. A cosieve on an object $c$ is a set of morphisms of domain $c$ and stable by postcomposition. Let $F:C^o\to Set$ be the presheaf mapping an object $c$ to the set of cosieves on $c$, such that for any morphism $f:c\to c'$ in $C$, a cosieve $S$ on $c'$ is mapped to the cosieve $\{h \circ f | h\in S\}$ on $c$. This $F$ is what I call the presheaf of cosieves.