We know that if two maps $\phi,\psi : \mathcal{F}\to \mathcal{G}$, where $\mathcal{F} $ is a presheaf and $ \mathcal{G}$ is a sheaf, agree on stalks then they are equal. Can we find an example to show that the condition $ \mathcal{G}$ be a sheaf is necessary and that merely assuming that it is a presheaf does not suffice ?
P.S. I am learning elementary sheaf theory and would find it very helpful if someone could suggest a good place where I can find lots of examples and counterexamples. Thanks