How do I know when my download actually finishes on PS4?

I got ACIV on the Playstation Store, the file was apparently 20GB but it only downloaded 4GB (according to Notifications/Downloads). Then I was able to access the game and play the story for a little while, then when I went to multiplayer, a message appeared saying that it was still downloading (about 60%). So my question is how do I know when all of the game download(s) finish because maybe the rest of the story is also downloading and there's no download on notifications/downloads (maybe there's another download screen that I don't know exists). Also do I need to start the game for the download to finish?

Only the initial install required before you can play the game is shown in notifications. To see the remaining, go to the game in the menu and press Options button then Information. It will show the download status.

You do not need to start the game for the download to finish, it runs in the background while the system is on (or in standby if you have enabled downloads in standby), you can even start another game and let the first game finish downloading in the background.

Thank you!