If you are looking to get the Caller ID and other native features from the device to work you need to export from Handheld Contact to the Contacts app on the device. 

For more info, click HERE for Android or HERE for iPhone/iPad.

However, there are only certain fields that get exported during this process.  They are:

iPhone:

MobilePhone
Phone
Home_Phone
LastName
FirstName
Company
PrimaryEmailAddress
Addr1, addr2
City
State
ZIP
Country

Note: After the 'Export' is complete the following fields in HHC will map to the iPhone Contacts fields:

HHC Field  -> iPhone Contacts Field

MobilePhone -> Mobile

Phone(Business/Work) -> Work

Home_Phone -> Main

PrimaryEmailAddress(Email) -> Work Email

Android:

Name (First and Last)
Phone
Mobile phone
Home phone
Email
Personal email
Company
Title
Business address (addr1 + addr2 + addr3, city, zip, state, country )
Home address (home address, home city, home zip, home state, home country )