Is the cokernel of a map of sheaves a seperated presheaf?

Question

The cokernel of a map of sheaves is not necessarily a sheaf until you sheafify. In every example I have seen of the cokernel failing to be a sheaf it is the glueability axiom that fails while the identity axiom still holds. My question is whether the identity axiom can fail and if so is there a simple example which demonstrate this?

Answer 1

No. If $X$ and $Y$ are two sections of the cokernel, then $X$ and $Y$ are images

Not always. Let $\mathcal F_1 \to \mathcal F_2$ be a map of sheaves where the presheaf cokernel does not satisfy the gluing axiom. Let $\mathcal F_3$ be the sheaf cokernel of this map. Let $\mathcal G$ be the presheaf cokernel of $\mathcal F_2 \to \mathcal F_3$. I claim $\mathcal G$ is not separated.

Indeed, every section of $\mathcal F_3$ is by definition locally a section of $\mathcal F_2$, so its image in $\mathcal G$ is locally equal to $0$. So take a section of $\mathcal F_3$ created by gluing that does not exist in the presheaf cokernel of $\mathcal F_1 \to \mathcal F_2$. It is not in the image of $\mathcal F_2$, so it gives a nonzero section of $\mathcal G$. So we have a nonzero section that is locally zero. Hence $\mathcal G$ is not separated.