If you mouse over the description of a pet in the inventory screen, the information screen shows all sorts of information on the pets stats and powers, but it does not show the name of the pet ability like "Curse Aura" or "Ghost Wail 1". To see the name of the pet ability, you need to go into the petrinarian screens.
Also, pet cooldown times does not show up anywhere and possibly this could be shown here too. However, this is not shown anywhere so possibly this is by design.