caught a cat

hello! i just caught a cat that’s been lurking around my work. ive been feeding them since i’ve seen them. he or she has been living in the drain pipes with its family (i think). i’ve only seen two cats so i’m not sure if there’s more or not. i caught the little critter last night, it’s been freezing cold recently ( it snowed last night) and it’s been making me so sad seeing he or she running around while it’s below freezing. he or she is absolutely horrified, i have them in my bathroom to keep them contained ( i haven’t taken them to the vet yet, going to try today) i just feel horrible, i feel like i created more stress for them. the look in their eyes makes me feel like it might have been better off with its family. maybe i’m feeling too guilty i just cant with the fear in its eyes, it makes me feel like a monster. i’m not sure if they are feral, most likely. i stuck my hand in the cage ( with gloves) to give it food, water and a tiny litter box and they just hid in the corner in fear and didn’t get aggressive. they are just so horrified. i wanna keep it and give it the best life it can have, they seem so sweet and timid. i’m just worried i’m not gonna be able to. any advice or reassurance would be greatly appreciated. i’ve been feeling sick to my stomach that i’m causing such a small little critter so much stress.