Let $F$ be a presheaf on a space $X$ and say that $F$ has property MY if for all $U, V$ open in $X$ we have an exact sequence $$0 \to F(U \cup V) \to F(U) \oplus F(V) \to F(U \cap V) $$
Is this property the same as the sheaf property ?
I think I can show it by recurrence.