Over the past decades, optimal transport theory has found broad applications across both applied and theoretical fields. A central role is played by the so-called fundamental theorem of optimal transport, which provides a precise link between primal and dual optimizers. Recently, nonlinear extensions such as entropically regularized optimal transport and martingale optimal transport have gained significant popularity. Gozlan, Roberto, Samson, and Tetali developed weak transport theory as a sufficiently general framework to formulate these problems within a coherent setting. We show that the principal results of optimal transport, such as existence, duality, and specifically the fundamental theorem, extend to the nonlinear case. In particular, we provide a unified derivation for the optimizers in the entropic transport problem, the Brenier-Strassen theorem, and the Martingale version of the Benamou-Brenier problem.