We study the structure of charged colloidal suspensions under confinement and determine a state diagram for the occurrence of electrostatic adsorption onto the confining walls, an effect that results in the accumulation of particles on the bounding surfaces and that could be relevant in experiments. We use Monte Carlo simulations to quantify this structural transition and perform theoretical calculations based on integral equations. Overall, our results provide a guide for experimentalists dealing with charged colloidal systems to determine the relevance of this purely electrostatic effect.