In this talk, we insist on the concept of nonnegative cross-curvature and its synthetic definition for a general cost on a product space. Then, by using a formal argument we show why one can expect that such a property should be also true for the Wasserstein space. Then, we give examples of cost satisfying this synthetic nonnegative cross-curvature, in particular a new one with the Bures-Wasserstein case. We extend the result to the case of unbalanced optimal transport and show some potential applications.