You can use the option in the attachment to move the library back to wherever you want it. Then go to Windows Explorer, select C:\Users\Patrick\Calibre Library and either press Ctrl+C or right-click and select Copy. Then either right-click and paste or use Ctrl+V to paste into The Dropbox folder.
Do NOT drag it there, that will only move it again!
However, first consider whether this is necessary. calibre can function just as well in either location, and if you only copy the library, then dropbox will not update as you add new books, since per my explanation, your doc was never opened from dropbox, only from My Documents.
Do you just want to be able to open the calibre library directly from your user folder, instead of going to dropbox first? If so, you may want to take a look at this link:
http://www.addictivetips.com/windows...out-moving-it/
You can create a symlink, so that the same folder can be opened from either location, and it is THE SAME FOLDER!!