Tried to repeat the Tutorial on adding the Save and Load system to the project. Saves components are added manually, no problems here. But when I try to click Auto-add Save components to GameObjects as in the video, absolutely nothing happens! The console shows exactly 0 changes. No components of objects for saving, in auto-add mode are NOT created!

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Check that your scene added to the Build Settings. This operation is a game-wide process, it's not limited to the specific scene that's currently open.
A notice to alert the user in this situation would be helpful, I think. Thanks for raising the issue, if this was the case - I'll look into adding this in the next update.
@ChrisIceBox Oh, it's really my inattention! I was starting the Tutorial over from 0 with new files, and somehow missed that moment with adding a scene in Build Settings. I added it and of course everything worked as it should. Thank you!