How do I make the camera stay on a fixed Z-pos where it moves X/Y delayed from the player object?

I am trying to make a player object that moves forward (or sideways) and to give the sense of "smooth movement" I am trying to damper the camera movements so the cameras position follows the main object, but delayed.

Both objects. Main/player object and camera have their own Z-pos. When player goes up (Y) or left (X) the camera already keeps the player in focus with smoothfollow target.

But I am trying to get the camera to move in "behind" the player after a while, sort of damping within 1-2 seconds.

That is, if we go in a streight forward, the camera positions itself behind player over time.

Initial positions:

left ----------------- player x-pos ----------------- right

           ^ LookAt(main) ^

.....then player moves left on X-axis:

left ------ player x-pos ---------------------------- right

           ^ LookAt(main) ^

after a second or something camera follows:

left ------ player x-pos ---------------------------- right

      ^ LookAt(main) ^

I have the camera following with dampered LookAt now, but I cant figure out which formular to make the camera position dampered too?

Any suggestions?

I am already using this for LookAtTarget

 // Look at and dampen the rotation
 var rotation = Quaternion.LookRotation(target.position - transform.position);
 transform.rotation = Quaternion.Slerp(transform.rotation, rotation, Time.deltaTime * damping);