I mean, I know that a category consists a class of objects and a class of morphisms, satisfying some axioms. What confuses me is that, in set theory, we define a group to be a tuple $\langle G,\cdot\rangle$ where G is a set and · is an operation. One can rewrite the sentence "$\langle G,\cdot\rangle$ is a tuple, $G$ is a set and $\cdot$ is an operation" in extremely formal language, if necessary. But——if we build up the category theory based on the language of set theory——what exactly is a category? A conglomerate that contains a class of objects and a class of morphisms?
If we build the category theory completely independent of set theory, then how would the axioms be like? Are they based on first-order language? Could they be wrote down in some way as formal as those axioms of ZFC or NBG? So far I haven't seen such a formal definition on my textbooks.
By the way, since I knew the word "conglomerate", I've been wondering if we could define some sense "larger and larger" after set, class and conglomerate, and even use something like "transfinite definition". But I guess this question may be too stupid to ask separately so I just wrote it here.