When Mihawk first arrived at Kuraigana Island, it wasn't completely abandoned, which he soon figured out after going into the large and gothic castle.
After arriving on the island, Mihawk had seen it was pretty much abandoned and a good bit hidden from people, which was perfect to make as a home. The swordsman had soon found a large castle and entered it, though he noticed it was in pretty well condition on the inside more than out. Mihawk had explored most of the place, but sensing another presence in the area with his haki... Like someone was watching him.
This only made the hawk-eyed man get cautious, yet as soon as he placed a hand on his sword, the presence seemed to disappear. For the rest of the night the other had seemed to disappear and Mihawk had settled in, which didn't take much, just finding a room and sleeping in it.
What the swordsman didn't know was that the person was still there, and well aware where he was.
Once Mihawk had awakened to a soft shuffling sound, he hadn't expected to see a... large bird? That's what it seemed to be, a large bird perched at the edge of the bed he was sleeping in, but it's head was hidden in it's wings, like it was sleeping. That bird had the same presence from earlier.