Sure. So you know how Duckling club is about several Wizards from different Colledges?
Well, running the Library will be sort of like that except Mat will be trying to create a staff of halflings, dwarves and humans, to run the Library, while also choosing stuff like how it's built, what the security is, negotiating contracts for new or ancient books, setting up terms and conditions for visiting scholars, universities, organizing books exchanges and so on.
Some of that will quite possibly involve books on otherwise forbidden subjects, or magic, or religion, and will bring Mat in contact with other Wizards, Witches and various priests, not to mention the other Colleges.
So in five years, a Head Librarian section might involve decinding what kind of protective measures we want on our Restricted Section of Books, navigating some kind of personal conflict between a dwarven and a human staff member, progress/negotiation reports on book acquisition, and so on. In this respect, a love of languages and Mat's xenophilia will serve her well, as will that she is a Laconic Buerocrat. Oh, and we might also get some Elves/Dwarves/Dragons rarer stuff, eventually coming around to see what all the fuss is about.
What I'm going to be really interested in is how Mat deals with her oaths as a Wizard, and sets policy on books that might have been fully banned under the Empire. Not the evil ones, but stuff that's politically inconvenient. That's going to create intrigue and diplomacy opportunities.
Basically, the Library will have a tangential impact on all the other work Mat is doing, much like the EIC has. And every once in a while, it will feature front and center, while doing it's own thing in the wider world all the while.