Context. Plasma outflow from a gravitational potential well with cosmic rays and self-excited Alfvén waves with cooling and wave damping is studied in the hydrodynamics regime. Aims. We study outflows in the presence of cosmic ray and Alfvén waves including the effect of cooling and wave damping. We seek physically allowable steady-state subsonic-supersonic transonic solutions. Methods. We adopted a multi-fluid hydrodynamical model for the cosmic ray plasma system. Thermal plasma, cosmic rays, and self-excited Alfvén waves are treated as fluids. Interactions such as cosmic-ray streaming instability, cooling, and wave damping were fully taken into account. We considered one-dimensional geometry and explored steady-state solutions. The model is reduced to a set of ordinary differential equations, which we solved for subsonic-supersonic transonic solutions with given boundary conditions at the base of the gravitational potential well. Results. We find that physically allowable subsonic-supersonic transonic solutions exist for a wide range of parameters. We studied the three-fluid system (considering only forward-propagating Alfvén waves) in detail. We examined the cases with and without cosmic ray diffusion separately. Comparisons of solutions with and without cooling and with and without wave damping for the same set of boundary conditions (on density, pressures of thermal gas, cosmic rays and waves) are presented. We also present the interesting case of a four-fluid system (both forward-and backward-propagating Alfvén waves are included), highlighting the intriguing relation between different components.