I’ve just joined this forum so thought i would have a go at answering this.
I would use smoothdamp to slowly follow the mouse position, you can add Time.deltaTime to this.
This is what i would do, script is attached to the camera, hope it helps!
var look_sensitivity : float = 1.0;
var curr_x_rotation : float;
var curr_y_rotation : float;
var look_SmoothDamp : float = 0.1;
var y_rotation : float;
var x_rotation : float;
//get desired rotation from mouse
y_rotation += Input.GetAxis("Mouse X") * look_sensitivity * Time.deltaTime;
x_rotation -= Input.GetAxis("Mouse Y") * look_sensitivity * Time.deltaTime;
//limit the x rotation (so you can't spin)
x_rotation = Mathf.Clamp( x_rotation, -90, 90);
//smooth the transition to desired rotation
curr_x_rotation = Mathf.SmoothDamp( curr_x_rotation, x_rotation, x_rotationV, look_SmoothDamp);
curr_y_rotation = Mathf.SmoothDamp( curr_y_rotation, y_rotation, y_rotationV, look_SmoothDamp);
//apply rotation to camera
transform.rotation = Quaternion.Euler( curr_x_rotation, curr_y_rotation, 0);