While category of sets technically could not be expressed as a ZFC set, the idea behind the set theory is enough. Also you could add an axiom[0] in ZFC to make

While category of sets technically could not be expressed as a ZFC set, the idea behind the set theory is enough. Also you could add an axiom[0] in ZFC to make category of set a set. [0]: https://en.wikipedia.org/wiki/Grothendieck_universe https://en.wikipedia.org/wiki/Grothendieck_universe