I want the player camera to move depending on where the player is on the screen

If you’ve ever played shadow of the colossus, you know what I mean. Either the player is at the left side of the screen, or the right, depending on their position. They are only ever center when they are climbing. The camera moves automatically. How would one approach doing this?

Mmmmh what about deciding the zone where the player must be on the screen, and using projection/deprojection look wherethe camera should move?