Local Demon Bestiary
Once accepted, head north of the Shinjuku Hunter's Assocation. Confirm to the impeding man that you are taking on the job and you'll be allowed outside.
East Entrance: Main Street
Begin by going east and south to find a Chakra Drop , then southwest to find another Chakra Drop . Head to the west, then north along the western wall to find a Dis-Poison . If you head far, far east of there, if you pay attention to the southern wall once reached, you can find a 1-Star and 2-Star Cosmetics .
If you head to the north of there and examine the wall angled northwest/southeast-like, you can find a demon's Domain. Within, go along the path and left at the fork for a Revival Bead . Backtrack and go straight, then left, then right to the door to the demon boss. Inside is the Level 27 Sui-Ki.
It is not particularly strong (around 40 ~ 60 damage per attack) and around 1,300 HP, is weak to Fire, and reflects Ice. Blaze it down for an easy win. After the Domain dissipates, you'll get a Corked Bottle , used in a Challenge Quest later.
From there, head way and once you return into contact with the buildings, keep an eye off to the north to find another relic, 1-Star Cosmetics . Further along this wall, you'll be able to find a Laptop PC in the northwest corner, used to fulfill one of Hugo's delivery quests from earlier.
Meanwhile, fight the nearby Corpse. Afterwards, backtrack along the wall to the open area and exit north to the next area.