Caffarelli's contraction theorem is a pivotal result in the field of optimal transport, providing a Lipschitz bound on the transport map, that is, a Hessian bound on the potentials, between log-concave densities. In this talk, I will present a paper by S. Chewi and A.A. Pooladian that extends this theorem to the entropic optimal transport problem. This generalization is achieved through the application of classical functional inequalities, marking a distinct approach from Caffarelli's original proof, which utilized the maximum principle. If time allows, we will explore how this method facilitates the derivation of second-order bounds for the Fokker-Planck JKO scheme across the entire space.