Haystack can automatically synchronize Haystack card data to your phone contacts (not the other way around).
Whenever changes are made to a Haystack card, the new details are automatically updated in your phone contacts.
This feature is normally turned ON by default. you can turn it ON/OFF in the Haystack's app 'settings' screen (view the settings screen by tapping the 'three horizontal lines' icon on the top right corner).
Note: If you do not want your Haystack cards to sync to your phone's contacts, you can uncheck this option in settings. Doing this will remove all Haystack card details from your phone contacts.
Note: The app does not sync your phone's contact into Haystack.