Ok, here is my critique. Overall, pretty good. Below are some re-writes (mostly word choices):
Book of Knowledge: When you borrow this book, you gain proficiency in two of the following skills of your choice: Arcana, History, Nature or Religion. While you have this book on your person, you have advantage to all ability checks where you can add your proficiency bonus from either of the two skill you selected.
Book of the Mage: When you borrow this book, select one of the following cantrips: Dancing Lights, Mending, or Message. While you have this book on your person, you know the selected cantrip and it is a Warlock cantrip for you. This cantrip does not count against your number of cantrips known
(N.B.: Theses are the "Wizard non-Warlock cantrips": Acid Splash, Control Flames, Dancing Lights, Fire Bolt, Gust, Mending, Message, Mold Earth, Ray of Frist, Shape Water, Shocking Grasp. I would select from those so the selection won't overlap with cantrips the Warlock can already choose.)
Book of Skills: When you borrow this book, you gain proficiency in one skill or tool of your choice. While you have this book on your person, your proficiency bonus is doubled for any ability check you make that uses the gained proficiency.
Book of Tongues: When you borrow this book, select a number of languages equal to your proficiency bonus. While you have this book on your person, you understand and can speak the chosen languages.
Book of War: When you borrow this book, you gain proficiency in medium armor and shields. In addition, you gain proficiency in two martial weapons of your choice.
Fervent Reader
As your relationship with the Living Library grows, it grants you additional access to its content. Starting at 6th level, you can borrow one additional book when you perform the ceremony at the end of a long rest. You can select the same book twice. If you already have borrowed two books and borrow a third one, you choose which of the previous two are returned to the Library. Furthermore, you can now select the following book:
Book of Spells: When borrowing this book, select two Warlock spells. While you have this book on your person, you know the selected spells and they do not count against your number of spells known.
Borrow Energy
You can now tap into the incredible arcane reserves of the Living Library. Starting at 10th level, you can spend 1 minute to contact the Living Library to regain one expended Warlock spell slot. Once you’ve used this ability, you cannot do so again until you finish a short or long rest.