Various forms of geoengineering have been proposed to counter anthropogenic climate change. Methods which aim to modify the Earth's energy balance by reducing insolation are often subsumed under the term Solar Radiation Management (SRM). Here, we present results of a standard SRM modelling experiment in which the incoming solar irradiance is reduced to offset the global mean warming induced by a quadrupling of atmospheric carbon dioxide. For the first time in an atmosphere-ocean coupled climate model, we include atmospheric composition feedbacks such as ozone changes under this scenario. Including the composition changes, we find large reductions in surface UV-B irradiance, with implications for vitamin D production, and increases in surface ozone concentrations, both of which could be important for human health. We highlight that both tropospheric and stratospheric ozone changes should be considered in the assessment of any SRM scheme, due to their important roles in regulating UV exposure and air quality.