In handover case, UE context is already created at target eNodeB, that's why Target cell sends the allocated C-RNTI to source and source cell in turn sends to UE in mobility control info IE.
When UE does RAR with the assigned preamble target eNodeB MAC maps the UE context and since all the contexts have already been created at target eNodeB handover becomes faster.
This is my understanding and interpretation of sending target eNodeB's C-RNTI in mobility control info.