[Snake]

It's not to prove a point, it's for environment consistency. The tent icon we see in the overworld is just a symbol of 'hey rememedia's tent!' or a straight up "This is the Arena. Why should we have the icon of a castle?", not what it actually is.

Also, whatever we do in the Arena in regards of Reputation counts as if it's Dormeho area, apparently? Is it intentional? Shouldn't that be like Cellsvich or Voilegard?