Hmmm.... I thought Ω was an object with just two points.

But then if Ω can have more than two points in some logic, then the definition of ∧ does not seem complete, because it doesn't explain how to choose mappings other than (true,true)→true.

If it's only two points in Ω, there is no choice; but if it's more points, then the specified definition doesn't seem to make ∧ a unique arrow.