Au idea where sometime after the great flood, Heaven decided to designate an official Ambassador to act at the liaison between them and Hell both to try and better curb the rampant conflict between the two realms and to act as the arbiter of the laws that would be set in place regarding how much and what kind of influence the two realms can have with humanity and earth. Adam is the one chosen to take on this task.
The first couple of centuries mostly involved setting up the laws and regulations, and Adam becoming acquainted with each of Hell's rulers, save Lucifer and Lilith who use intermediaries as they have avoided being in the same vicinity as the First Man. While at first the Sins care little for Adam, each begins to grow close to him as he takes the time to get to know them, their style of ruling, and learning to accommodate their personal sin that they embody when interacting with them. Lucifer and Lilith are frustrated by the Sins growing closer to Adam due to their own biased memory of what went down in Eden. They spend a few millennia avoiding any contact with the First Man, which suits Adam just fine as he is content to never have to interact with the two people who gave him is first heartbreak, as he had loved them both back during their time in Eden.
Turns out Adam is pretty good at his job, and is also pretty impartial as despite being an archangel, he is not afraid of laying down the law to Heaven when they try any funny business. Heaven might be more orderly and just be it doesn't mean they can't be difficult and overstep. This definitely puts a strain on his personal relationships with the higher angels of Heaven but Sera generally trusts and respects Adam's judgement even if she disagrees.
The Heavenly Embassy would be his primary workplace, and it actually exists in all seven layers of Hell at once. Adam's embassy staff are also demons, since no angel wants to work full time in hell. I'm seeing him having a sweet little imp assistant that he treats like a daughter. It's actually considered a high honor to work for the Ambassador, and is also one of the safer jobs in Hell since Adam has shown to be fully willing to go after anyone who attacks or harms his staff or their families.
I can see that if this get to the time or starts at the time where the canon takes place, then the yearly Extermination were recently set into motion. However, this was proposed and planned by the Archangels with Sera's permission instead of being Adam's idea. The Ambassador is put in charge of carrying out the Exterminations and insuring the pact laying out the rules for the Exterminations are adhered to. He isn't a big fan of doing them but between the disastrous event a decade ago that caused Heaven to act in such a manner, Lilith and Lucifer refusing to ever meet with him to discuss anything, and the sinners becoming more and more of a problem in recent years, his hands are tied.
That's what I got so far. I can see Adam ending up with one or more of the Sins, maybe even all of them. Could also see this ending up as either Adamsapple and/or Guitarlily in the long run as the two of them start to actually get to know Adam and end up deciding they want him to be tier third. Imagining them trying to court Adam and being really bad at it because they are "old" and out of practice and Adam just being confused by their antics. Could also see this being Guitarsins (personal ship name for Adam x 7 Sins) or even Adamsdemons (personal ship name for Adam x Lilith x 7 sins)