This is the 14062nd most frequent Korean word.
맺고서
"맺고서" means "after forming or concluding."
Here, "맺고서" is used to indicate the action of forming (협력을 맺다) an agreement and then doing the subsequent action (starting the project).
그들은 협력을 맺고서 프로젝트를 시작했다.
They formed a cooperative agreement and began the project.
In this sentence, "맺고서" shows the sequence where a contract (계약을 맺다) precedes the reservation of the room.
계약을 맺고서 방을 예약했다.
After signing the contract, they reserved the room.
This usage of "맺고서" emphasizes the establishment of a relationship (관계를 맺다) followed by the emotional response (feeling responsibility).