  In this paper, we provide an alternative proof of the monotonicity principle
for the optimal Skorokhod embedding problem established by Beiglb\"ock, Cox and
Huesmann. This principle presents a geometric characterization that reflects
the desired optimality properties of Skorokhod embeddings. Our proof is based
on the adaptation of the Monge-Kantorovich duality in our context together with
a delicate application of the optional cross-section theorem and a clever
conditioning argument.
