Oh sure, you can call them whatever you like. But if you have an ADT system which has a unit type that isn't quite the empty product then you'll be in for an in

Oh sure, you can call them whatever you like. But if you have an ADT system which has a unit type that isn't quite the empty product then you'll be in for an interesting time at least. E.g., this is true in Haskell and it causes there to be some "extra structure" which often has to be smoothed over. Lots of library functions exist just to do this smoothing.