A nightly visitor

For the past few months this stray cat has been a visitor to my yard. He showed up in October and was very skinny . ...I had to feed him of course, being the cat lover that I am. He came every night and would stand outside and meow if there was no food in the dish.. When winter set in , I was worried about him and noticed he was sleeping in the shed next door ( the place is empty due to foreclosure) I made him a box and put a blanket in it and made sure he had food.
As winter approached and it started to snow I put him in the house one night. My cat was not thrilled!!! He ate and wanted back out and so I put him out knowing he would find shelter.

Lately he has decided he would like to come in and will stay for hours and some times spends the night. Below is a picture of him sitting on the steps trying to decide if he wants in.

I have tried to find the owner and so far, no luck. I even tied a string and note around his neck thinking maybe he had another home...not the case though.

Here he sits in the window looking out and below he is nice and warm in the bed I bought for my cat years ago and she would not use it. I'm sure he must have belonged to someone as he is litter trained and very gentle even though he has claws.

Yes the two cats do share the house...mine just pretends he isn't here and spends her time sleeping on my bed . I'm sure he will spent time warming up before wanting back out again.