How to add contact to Messages on iPhone

If you want to see peoples names in the message app, you will have to add those persons to your Contact app first.

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1. Go to the Contacts app on your iPhone.
2. Tap +
3. Enter details for the contact such as name and phone number, then tap ✔️

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copied from: Add and use contact information on your iPhone - Apple Support

After the contact info has been added, you can fill in the name, instead of the phone number in the Message app, and the contact app will suggest a list of related names to select from.

Once the name is connected to a phone number, you will see the name listed in the conversation list in the message app and not the phone number.