Support

The only way to downgrade the Professional version to the Core version is to install the Core version on top of the Professional version. This CAN NOT be done automatically by Admin Tools or by any code we, Akeeba Ltd, have written.

There are, however, three ways this can happen that I can think of the top of my head:

1. A Super User on your site did it. They could accidentally install the Core version through the Install from Web tab in the Joomla extensions manager. Or they could upload and install the Core version. Joomla will not warn you when you are installing the Core version on top of the Pro version.
2. If you had previously installed the Core version it's possible that its Update Site (how Joomla finds updates to extensions) might have not been replaced, which is rather unlikely. Or, more likely, there might have been an available update to the Core version cached by Joomla. Please remember that updates to our software IS NOT managed by our code. It is managed by Joomla. Joomla uses a weird way of determining and installing updates. It queries the update site URL to figure out if there is an update available and stores the update information in the database. This is what you see in the Updates page of Joomla's extension manager. This information is also cached in a file for reasons unknown – a fix for this blatant bug is scheduled for Joomla 3.10 and came after more than 5 years of yours truly shouting my lungs out that it's a bug with an one-line fix, so why the heck is it still broken. But I digress. How Joomla currently works means that you CAN end up updating an extension you have upgraded to its Professional version or even an extension you have uninstalled in some cases.
3. If you are using a third party extension or service to install updates to your extensions it may have cached an update to the Core version or misidentified Admin Tools Professional for Admin Tools Core and applied an "update" to Core.

As you understand we do not have any input to or control over any of these ways which would let you overwrite Admin Tools Professional with Admin Tools Core.

Please refer to my previous reply. This IS NOT something that any code we have written can possibly do. We can only offer support about our own software. We cannot offer support about what you, your host or someone with access to your site has done, is doing or will be doing. This is completely outside the scope of our support as we objectively cannot help with.