Unity: transform.LookAt(target) not "looking at" target?

Question

I have been troubleshooting a prefab for several hours and am out of ideas. It concerns a method that I thought I was comfortable with, transform.LookAt(target). I have this prefab:

As you can see, I verified that the registration points for the eyes are at the center of mass; I also verified that they rotate correctly for the X, Y and Z axes. That yellow ball you see is the target object and IS NOT inside the prefab.

I have tried two means to get the eyes to look at the target. The first is this script, and it caused the eyes to dart 90 degrees counter-clockwise about the y axis (which is very wrong). I applied this script to both eyes. Check it out:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class LookAt : MonoBehaviour
{
    public Transform target = null;

    // Update is called once per frame
    void Update()
    {
        transform.LookAt(target);            
    }
}

And please, before you think it: Yes, I remembered to set the target object in the editor. You can see the eyes pointing to the side instead of the target:

I found this other script on the Unity documentation that didn't work at first, but then I changed "Rotation" to "LocalRotation", saw that I was close, reversed the "relativePos" vector, and it worked--but only about the y axis:

using System.Collections;
using System.Collections.Generic;
using UnityEngine;

public class LookAt : MonoBehaviour
{
    public Transform target = null;

    // Update is called once per frame
    void Update()
    {
        Vector3 relativePos = transform.position - target.position;

        // the second argument, upwards, defaults to Vector3.up
        Quaternion rotation = Quaternion.LookRotation(relativePos, Vector3.up;
        transform.localRotation = rotation;

    }
}

You can see here that it works in the XY plane at eye level, but take my word for it when I say the eyes cannot look up or down if I move the ball up or down:

How can I fix this so that the eyes always look at the target I need it to? Thank you sincerely for your time.

EDIT: I fixed it by this sloppy additional rotation, but there has to be a better way:

void Update()
    {
        //transform.LookAt(target);

        Vector3 relativePos =  target.position - transform.position;

        // the second argument, upwards, defaults to Vector3.up
        Quaternion rotation = Quaternion.LookRotation(relativePos, new Vector3(0,1,0));
        transform.rotation = rotation*Quaternion.Euler(0,90,0);            
    }

Answer 1

The Problem:
Your eyes default rotation makes them look to the negative X axis.
However, Transform.LookAt() rotates the transform, so that the Transform.forward looks at the target.
The Transform.forward always looks to the positive Z axis.
(All directions in local space).

Solution 1:
Rotate the texture or mesh, so that the eye's default look direction is along the positive Z axis.
Then LookAt will work as expected.

Solution 2:
After the call of Transform.LookAt, correct the rotation by the correct rotation.

transform.LookAt(target);
transform.rotation *= Quaternion.FromToRotation(Vector3.left, Vector3.forward);

Where Vector3.left represents the default forward direction of the eyes.