.223 is actually 5.66mm, but as far as I know, you can say .223 and 5.56 mm interchangibly. I think all the SS109/M855 .223 ammounition that the military uses is "officially" referred to as 5.56 NATO for the benefit of the rest of the world, I'd imagine.
Something I've always found interesting is that a lot of these calibers can be very different in what they are, yet very close in numerical diameter...
.357 is 9.06mm So, as far as diameter goes, .357 and 9mm are 60/1000ths different in diameter. A sheet of paper is about 100/1000ths thick. That being said, .357 Magnum and 9mm Parabellum are different bullets, of course.