We characterize the set of all obviously strategy-proof and onto social choice functions on the domain of single-peaked preferences. Since obvious strategy-proofness implies strategy-proofness, and the set of strategy-proof and onto social choice functions on this domain coincides with the class of generalized median voter schemes, we focus on this class. We identify a condition on generalized median voter schemes for which the following characterization holds. A generalized median voter scheme is obviously strategy-proof if and only if it satisfies the increasing intersection property. Our proof is constructive; for each generalized median voter scheme that satisfies the increasing intersection property we define an extensive game form that implements it in obviously dominant strategies.