A two-component Bose gas under rotation, with weak Coulomb or contact interaction is examined as the interaction between the different species is diminished. The larger component is seen to dominate the system, especially at weak inter-species interactions, such that the system appears to be a one-component system where the smaller component has at most the effect of a perturbation. On the other hand, when the two components have similar but yet unequal sizes, a smooth transition from a coreless vorex (typical for a two-component system) to a centered vortex in each component (typical for two independent one-component systems) occurs, as the interaction strength between the components decreases.