The nanowire platform offers great opportunities for improving the quality and range of applications of semiconductor quantum wells and dots. Here, we present the self-catalyzed growth of InAs/InSb/InAs axial heterostructured nanowires with a single defect-free InSb quantum dot, on Si substrates, by chemical beam epitaxy. A systematic variation of the growth parameters for the InAs top segment has been investigated and the resulting nanowire morphology analyzed. We found that the growth temperature strongly influences the axial and radial growth rates of the top InAs segment. As a consequence, we can reduce the InAs shell thickness around the InSb quantum dot by increasing the InAs growth temperature. Moreover, we observed that both axial and radial growth rates are enhanced by the As line pressure as long as the In droplet on the top of the nanowire is preserved. Finally, the time evolution of the diameter along the entire length of the nanowires allowed us to understand that there are two In diffusion paths contributing to the radial InAs growth and that the interplay of these two mechanisms together with the total length of the nanowires determine the final shape of the nanowires. This study provides insights in understanding the growth mechanisms of self-catalyzed InSb/InAs quantum dot nanowires, and our results can be extended also to the growth of other self-catalyzed heterostructured nanowires, providing useful guidelines for the realization of quantum structures with the desired morphology and properties.