Cat sitting, stray cat wont leave.

[DanHumphreyIsA]

I am catsitting for my nan, who also feeds some stray cats.
Today this sweet boy has wandered in (previously posted about another poorly cat, not the same one incase anyone saw that)

I've took food out, and he's got a cat tunnel thing in the garden to sleep in.
But he won't leave.
He's never been inside, and always skittish in fact this is the 2nd time I've seen him for a few weeks, so I'm not sure why he's had a change of heart today!

I can't take him to mine or obviously leave him here with Grem.
Should I just take him back outside with plenty of food? I feel awful sending him back out to the garden though, he's happily washing at the minute.

We live in an area sadly with so many neglected cats, so many are skinny, ragged

You can't tell from the photo but his body is quite skinny, and there is a bit of crusty skin around one of his eyes