A quantum-mechanical model description of a molecular photoswitch is developed. It takes into account (i) the electronic curve crossing arising from the cis-trans twisting of a double bond, resulting in an ultrafast internal-conversion process of the system and (ii) the coupling of the initially excited chromophore (the "system") to the remaining degrees of freedom (the "bath"), affecting a vibrational cooling of the hot photoproducts. The latter mechanism is responsible for the localization of the molecule in the cis and trans configuration, respectively, thus determining the quantum yield of the photoreaction. Following a discussion of the validity and the numerical implementation of the Redfield formulation employed, detailed numerical studies of the time-dependent dissipative photoisomerization dynamics are presented. While the short-time dynamics (less than or similar to1 ps) is dominated by the coherent wave-packet motion of the system, the time evolution at larger times mainly reflects the interaction between system and bath. The quantum yield of the cis-trans forward reaction (Yc -->t) and the trans-cis backward reaction (Yt -->c) is shown to depend on the energy storage of the photoreaction and, in particular, on the form of the system-bath coupling. On the other hand, it is found that Yt -->c=1-Yc -->t, that is the population probabilities of the cis and trans configuration at long times do not depend on the initial preparation of the system.