Arrgh, I wish to delete this, but do not know how. I cannot think of any answer as good as that of Georges. I was trying to think of a complementary one that violated the uniqueness axiom s1 on every neighborhood of some point. Help, Georges?

Arrgh, I wish to delete this, but do not know how. So i will make it into a question. Georges' nice answer is a presheaf that violates the existence axiom s2 on every nbhd of a point. Is there an example that violates the uniqueness axiom s1 on every nbhd of some point?

sheaves are ptesheaves which satisfy two additional axioms, called usually s1 and s2. failing s1 is enough to give a counterexample.

Arrgh, I wish to delete this, but do not know how. I cannot think of any answer as good as that of Georges. I was trying to think of a complementary one that violated the uniqueness axiom s1 on every neighborhood of some point. Help, Georges?