They call it the Tioga Library Building, I guess because it is connected to the historic but still not renovated Tioga Building (1890).
It is across the street from the Swiss Tavern and the skybridge that connects it to the University of Washington Tacomas existing library crosses over the Prairie Line. The official opening is Oct. 3. Fall quarter begins Sept. 24 and it looks like finishing touches and the final move in will be completed by then. It cost $24 million.