Archbishop Gomez seated on the LA Cathedra. Cardinal Mahony is to the left vested in the pallium. Cardinal William Levada is immediately to the right of Archbishop Gomez.
I am not sure this is liturgically correct. Usually, the Ordinary gets to possess his diocese during the Mass of Installation where he is given the crozier and is seated on his cathedra, obviously in his Cathedral.
I once read that a cardinal enjoys the right to sit in the cathedra of any bishop if he so decides to sit during the Mass which he celebrates. A papal legate also enjoys this privilege. A metropolitan archbishop, once he receives his pallium, meaning he has received his authority to govern over his metropolitan province, can sit on the cathedra of his suffragan diocese.
Only the pope can sit on the cathedra of any Roman Catholic bishop without seeking the permission of the bishop, accounting for his universal jurisdiction as shown on the photo below. The arms on the cathedra are those of New York Archbishop-Emeritus Edward Cardinal Egan
Here is the video I mentioned above.