Motivated by the dynamics of a continuous transformation defined on a compact metric space, wherein one studies the pressure of a real-valued continuous function and its relation to the Ruelle operator leading to a variational principle, we investigate the dynamics due to a holomorphic correspondence defined on the Riemann sphere and explore analogous concepts. In particular, we consider a Holder continuous potential defined on the support of the Dinh-Sibony measure for the considered correspondence and prove the uniqueness of its equilibrium state.