Generalized Pólya urns with non-linear feedback are an established model for the competition in markets. Depending on the feedback function, the model can exhibit monopoly, where a single agent achieves full market share. We examine the asymptotic behaviour with diverging initial market size for a large class of feedback functions, and establish a scaling limit for the evolution of market shares, including a functional central limit theorem. In the monopoly case find a criterion to predict the (in general random) monopolist with high probability under generic initial conditions. Our results reveal an interesting difference between exponentially and more realistic polynomially growing feedback.