Has you authorize his Disktop with the same Adobe ID?
I don't really know. I've made the installation 2 years ago on my laptop and can't remember.
Is it a middle to know the adobe Id used by my laptop's Digital Edition ? When I look the e-book's properties on bith lachines, I see the same e-mail address, but idon't know if this is a sufficient indication.
What make me thinck it's the same Adobe Id is that I see the book on both machines. But it's authorized on one and not on the other.
You should open Digital Edition on both devices. Then go to help > display Authorization Details. In this windows display the adobe id how has used for the authorization.
So it's the same e-mail address, and the same Adobe Id (urn:uuid: ....) on both devices.
But the versions of Digital editions are not the same : 4.5 on my laptop, and 4.5.2 on my desktop.
I haven't updated the version on my laptop, because it's the only device where I have access to my e-book, and I'm afraid to lose it on this device too !
The version of Digital Edition are seems not the problem because you have no access. Better you ask the reseller of the eBook with which adobe id the eBook are linked.