An eNodeB maintains UE old context including old C-RNTI as parameters.
After the RLF when UE triggers RRC Re-establishment Request message to eNodeB, it verifies the received IEs with the stored one.
Though C-RNTI term seems in the context the MAC layer but RRC layer or eNodeB application maintains UE context based on the C-RNTI as well. RRC layer or eNodeB application may need to find UE Context based on the C-RNTI in case of re-establishment procedure.