The objectwise cokernel presheaf can fail to be a sheaf

Let $\mathcal{P}$ be the presheaf cokernel of the inclusion $\underset{‾}{\mathbb{Z}}\hookrightarrow \mathcal{C}$, so $\mathcal{P}(V)=\mathcal{C}(V)\mathrm{/}\underset{‾}{\mathbb{Z}}(V)$ for each open $V\subseteq S^1$. Cover $S^1$ by the arcs $U_0=S^1\setminus \{(-1,0)\}$ and $U_1=S^1\setminus \{(1,0)\}$, and choose continuous angle functions ${\theta }_0:U_0\to (-1\mathrm{/}2,1\mathrm{/}2)$ and ${\theta }_1:U_1\to (0,1)$ with $e^{2\pi i{\theta }_i(z)}=z$. On $U_0\cap U_1$, the difference ${\theta }_1-{\theta }_0$ is locally constant integer-valued, so the classes $[{\theta }_0]\in \mathcal{P}(U_0)$ and $[{\theta }_1]\in \mathcal{P}(U_1)$ agree on the overlap.

If these local classes came from a global class $[g]\in \mathcal{P}(S^1)$, then on each arc $U_i$ the difference $g-{\theta }_i$ would be integer-valued and continuous, hence constant because $U_i$ is connected. On the two connected components of $U_0\cap U_1$, this would force the same constant difference to be both $0$ and $1$, which is impossible. Therefore the compatible local classes of step 1.1 do not glue in the presheaf $\mathcal{P}$.

So the objectwise cokernel presheaf is not a sheaf. By [F1], this is exactly why one sheafifies it to obtain the true cokernel sheaf. The statement is false.