What kind of institution is Gotei 13? Authority? Justice? Policy maker? According to Wiki:


  According to Yhwach, the original Gotei 13 were "defenders" in name only, comprising nothing less than a brutal mob of killers, but it was for this very reason they were a force to be feared, especially because Yamamoto was a ruthlessly pragmatic leader whom would hold even his subordinates as expendable. It is noted the Gotei 13 underwent dramatic changes after the extermination of the Quincy, mellowing out in the ensuing peace, having now found a sense of justice upon discovering things to protect and treasure.


Does this  mean Gotei 13 is acting as private organisation in Soul Society or is it an authority or law maker than can judge souls. If Gotei 13 is govermental authority, How can they exterminate Quincy if they act on behalf of Soul King? I know Soul King doesn't directly control Soul Society. And I believe what Gotei 13 has done was unjust. Therefore, I conclude that Gotei 13 is not ultimate source of justice. Therefore,  every soul can do whatever it wants in Human World, Soul Society, or Hueco Mundo without disobeying anything since authoritive figure doesnt exist among souls. 

I believe the only thing that makes Gotei 13 an authoritative institution is because they have group of Bankai users that are willing to punish other soul on behalf of Yamamoto.
The Gotei 13 was mainly used to keep the control and order in the Soul Society. They are also the first line of defense from any enemies, such as Hollows, Quincies, etc. They maintain the balance between the spirit world and the real world by eliminating Hollows, and guiding lost souls. 

The main purpose of the 13 was to provide an authoritative sense. Hope this answers your question.