Here is the full list of all 15 Qurare: Magic Library trophies - there are 13 bronze, 1 silver and 1 gold.
Extract Super Rare or higher Kodex x 100.
Consume 1,000,000 Gold.
Reach Lv. 70 with Super Rare or higher Kodex x 10.
Interpret Super Rare or higher Kodex x 20.
Clear all chapters in the Main Story.
Reach Fellow Lv. 70.
Complete 100 Co-op Mode battles.
Bind Super Rare or higher Kodex x 20.
Combine Kodex x 50.
Reach librarian Lv. 70.
Set up your deck with Super Rare or higher Kodices.
Clear White Day in the Main Story.
Obtain Main Story Kodex x 100.
Set up your deck with Rare+ or higher Kodices.
Collect all the librarians.