There are so many places, in arena and especially war, where one needs to be able to see what hero class a given hero has assigned. But in the current release of the game, you can only see hero class in the line up screen where you have chosen to fight the line. In arena, this requires you to not be on cooldown and have available hits. In war, this requires you to have enough points to attack the line you want to inspect.
In war, the additional problem arises when looking at the logs and trying to understand the full picture of a fight. There is no way to see hero classes in this UI.
My suggestion is to add a hover UI when you (long) press on the hero portraits of the line. In this popup, you could include all kinds of information that doesn’t fit in the very small real estate available for the mini-portraits.