Recommend you start with the product documentation about licensing:
http://www.cisco.com/en/US/docs/voice_ip_comm/cucm/admin/7_1_2/ccmsys/a02licen.html#wp1139268
Then check this:
Release Notes for Cisco Unified Communications Manager Release 6.1(3)
http://www.cisco.com/en/US/docs/voice_ip_comm/cucm/rel_notes/6_1_3/cucm-rel_note-613.html#wp472518
Installation/Upgrade (Migration) Considerations
After you upgrade to Cisco Unified Communications Manager 6.1(3) from a compatible Cisco Unified CM 5.x or 6.x release, the Cisco CallManager service does not automatically run, even though Cisco Unified Serviceability shows that the Cisco CallManager service is activated.
Same aplies to 6.1.4
If you have SW Feature license for 6.0 ( a real license, not a demo license), then it will work on 6.1.4 as well, on upgrade.
Please make sure the customer is operating on a real 6.0 SW Feature license and then upgrade.
To check whether it is a "real" SW Feature license please look at the uploaded license files on the License Upload Page.
HTH
Gonzalo