No updating ADE library between two PC

The last ebook I downloaded taken as a loan from a public library was  not updated  on my  notebook (OS  Windows 10 ) regularly  authorized by ADE  4.5.2  with Adobe  ID . On the other side, the same ebook is opened and displayed rightly on my desktop authorized  with the same  Adobe ID.What have I to do  to solve the problem? Thanks a lot for any useful advice. 

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