Dual mapping theories (e.g., Atkinson, Bower, & Crothers, 1965; Lacouture & Marley, 1991, 1995) propose that psychophysical judgment involves two component processes: stimulus representation and response selection. This empirical and theoretical paper studies the relative contributions of these two component processes in determining performance in the identification and categorisation of unidimensional stimuli. We extend Lacouture and Marley's (1995) mapping model of absolute identification to categorisation, and demonstrate that the extension predicts that the number of response categories has a much larger effect than the number of stimuli in determining various aspects of the relevant data, including the so-called bow effect. This prediction is supported by the results of three experiments in which the number of stimuli and/or the number of response categories was manipulated.