We criticize some conceptual weaknesses in the recent literature on coalitional TUgames and propose, based on our critics, a new definition of dual TU-games that coincides with the one in the literature on the class of super-additive games. We justify our new definition in four alternative ways: 1. Via an adequate definition of efficient payoff vectors. 2. Via a modification of the Bondareva-Shapley duality. 3. Via an explicit consideration of "coalition building". 4. Via associating general TU-games to coalition-production economies. Rather than imputations, we base our analysis on a modification of aspirations.