The battle system used in the game is pretty simple and straightforward. You have advantage over certain elements, so it's better to build teams taking this into account.
Special multipliers Edit
This bonus last two weeks (14 days), and starts when the card is on sale at the daily summons. It doesn't matter if you already had the card and don't buy it, the servant will get the bonus anyway.
The multiplier acts before calculating the elemental advantage.
You can choose two ways to battle, with a small Team or a Company.
The first method is the most common and used. Here you have three teams you can arrange, each one with three Sidians. You can arrange them in the order you want and also you can have teams organized how you want.
When creating your teams, first you need to consider their combined AK and HP, and then you also need to take into account skill, element, and AK multiplier.
With the second method you can arrange a company that contains up to 9 sidians, which will have obviously higher AK and HP that a normal team, but will also cost you more MP. Having 1-3 servants will cost you the same as a normal team, 4-6 will cost you double and 7-9 will cost you triple.
This method is only useful when fighting story bosses, since you have to beat them in one go and that can be difficult to do at lower levels.
Companies can be classified by a grade letter, which will be given based on the total AK if this is the higher, or an average between HP and AK if HP is higher (Unconfirmed, I just tried about 20 tests but may change later, not that it's really important though)
After battle, if you beat the Sidian you're fighting with, you will get one card and sometimes a special drop item, like Shining Powder or Runestones.
When fighting Rare irregulars, there's a low probability of getting the Sidian you beat, and if you don't, you'll get a random Normal or Uncommon card (and in rare cases, a random Rare card).
When fighting Exceptionals, whatever the rarity, you have a low probability of getting the Sidian you beat, and are probably going to get a random Normal, Uncommon or Rare card.