Just a word of caution.
Cats can be extremely territorial and (quite obviously when you think about it) don't have a concept of human 'ownership'. They are instinctively 'pack' or 'litter' orientated. As such, if you show a stray cat too much acceptance, it can have an extremely detrimental effect on your resident moggie. Completely outside of your control is the scope of feline assertion; its doesn't matter if your own cat (who I assume you are more emotionally attached to) is bigger, smaller, older or younger, if the stray has the 'personality' to assert dominance, it will, especially if the outsider has found a ready source of food and relative safety. Strays, in my experience, seem rather adept at asserting dominance.
Yonks ago we had two cats, and as a tag team they were pretty formidable in the neighbourhood. But as a family of cat lovers, when a grey cat (who we named Smokey) appeared on the scene, looking pretty pathetic, thin and scrawly, we took pity and started feeding it. Initially it was very timid and passive, we took further pity on it and given its appearance, we simply couldn't let it go hungry. However, within a few months, it had grown fat, comfortable, and the garden became a practical 'no-go' area for our own cats as Smokey waited around idly for the next feeding, and then rather nonchalantly bullied our own cats in their own kitchen for first eats at food time.
This was rather distressing to us, and eventually we solved it with threat displays towards Smokey (strictly non-violent, we love cats, but had to send a 'pack' message), but it took us a while. It was sad. We had almost come to love Smokey, but the dynamics wouldn't allow a happy litter, and after we scared it off we never saw it again. We didn't know its fate, but ultimately, it wasn't our cat, and we already had our own 'litter' to look out for.
I would personally say, if you love/respect your own cat, keep they stray at arms length. If you leave it any compassion food, leave it minimal amounts at the furthest distance from your property. Phone the RSPCA or Blue Cross to see if they can take it away and home it. If you let it get greedy and close, it can really upset your own cat and it can be quite a difficult situation to resolve sensibly.
And you cant sit cats down to explain the situation to them,