Constant Presheaf not necessarily a sheaf. Proof?

The argument in the Wikipedia article states that you should take the empty covering of the empty set, i.e. your indexing set (where $i$ comes from) is the empty set. The empty union is the empty set, so it is a covering.

Then you pick any $s \neq s^\prime \in A$ and consider them to be sections over the empty set ($A$ needs to have at least two elements). The condition that they agree after restricting is an empty one (because there is no set to restrict to), so by the local identity axiom, you would need $s=s^\prime$, a contradiction.