An Evening House Guest

My mom went downstairs a few nights ago, and saw a white face in the darkness of our front room, that then proceeded to jump out the one window that is usually left open.  She then remembered there had been a cat that had been begging at the back door, that now apparently likes to sleep in our comfy chairs at night (now i see the gray fur in the chairs).  So last night, the window got shut by the wind and it was stuck in the house.  It hid in the bedroom for a while and then ran out the front door after a few minutes of not coaxing it. I think the cat wants to adopt us.