3rd person camera doesn't rotate as intended

Hi folks; I'm having a bit of a problem with my 3rd person camera setup. I downloaded a reference file for the camera from the internet which lacked up-and-down movement, so I had to code that in.

However, when I now move up or down and then left or right, the camera moves in unexpected directions. I included a unitypackage file so you can see for yourself: http://www.mediafire.com/download/y9syj4udg1gk5ll/example.unitypackage

And here's a copy of the code used for the camera:

 using UnityEngine;
 using System.Collections;
 
 public class MouseAimCamera : MonoBehaviour {
     public GameObject target;
     public float rotateSpeed = 6;
     Vector3 offset;
 
     void Start() {
         offset = target.transform.position - transform.position;
     }
     
     void Update() {
         float horizontal = Input.GetAxis("Mouse X") * rotateSpeed;
         target.transform.Rotate(0, horizontal, 0);
         
         float vertical = Input.GetAxis("Mouse Y") * rotateSpeed;
         target.transform.Rotate(vertical, 0, 0);
     
         float desiredAngleX = target.transform.eulerAngles.y;
         float desiredAngleY = target.transform.eulerAngles.x;
         Quaternion rotation = Quaternion.Euler(desiredAngleY, desiredAngleX, 0);
         transform.position = target.transform.position - (rotation * offset);
         
         transform.LookAt(target.transform);
     }
 }

If you have any idea how to fix this, please let me know!

Cheers,

• Bageras