This paper studies duopoly in which two-sided platforms compete in differentiated products in a two-sided market. Direct competition on both sides leads to results that depart from much of the current literature. Under some conditions the unique equilibrium in pure strategies can be computed. It features discounts on one side and muted differentiation as the cross-market externality intensifies competition. Less standard, that equilibrium fails to exist when the externality is too powerful (that side becomes too lucrative). A mixed-strategy equilibrium always exists and is characterized. These results are robust to variations in the extensive form. The model may find applications in the media, internet trading platforms, search engine competition, social media or even health insurance (HMO/PPO).