This is a continuation of our accompanying paper [SIAM T. Control Optim., 54 (2016), pp. 2174-2201] We provide an alternative proof of the monotonicity principle for the optimal Skorokhod embedding problem established in [Beiglbock, Cox, and Huesmann, Optimal Transport and Skorokhod Embedding, preprint, 2013]. Our proof is based on the adaptation of the Monge Kantorovich duality in our context, a delicate application of the optional cross-section theorem, and a clever conditioning argument introduced in [Beiglbock, Cox, and Huesmann, Optimal Transport and Skorokhod Embedding, preprint, 2013].