Din also makes sure they look like they're together, keeping stride with the other man. If need be, he'd say Obi was visiting his Dominant for possible contract. They practically encourage that, so might work. Plus helping with that point, Din leads them to the Up housing, and to the familiar 012-A. Which thankfully, for Obi-wan, has heat.

"Make yourself comfortable." He says. Luke's not home, must be out at work, or something. So it's just them for the time being.