"Hrm, I'll just go into basics then," Yamato said briskly, "You have different types of contracts when it comes to demons. You can have temporary ones to an individual, or a cause - this is the type of contract Kujou made with Shiva. They're usually done for a single act, and require a form of... tribute. These are most common with more powerful demons. The second contract is something you're more familiar with - ones that are made until death of the human. Once you die, the contract is null and void. Well, usually, in any case."

His contract with Cerberus was still in effect despite dying in his world - perhaps his revival remade the contract? This never happened before so... maybe Yamato had unknowingly taken advantage in a loophole.

"And lastly, the third contract is made between a demon, or entity, and a clan. So long as the clan exists, the creature would be bound to serve them until every last drop of that bloodline is extinguished," Yamato smiled wryly, "Not many sign this contract, for obvious reasons, but this is the type of contract the Hotsuin have with the Dragon Stream. By using this contract, I can use it as a type of... how to explain... I can use it to call forth the Dragon Stream into this world. If all goes well, of course. There are many factors that can make this fail, as the Dragon Stream is a special case..."