I used some results from the category theory without thinking about its foundations. However, after reading a few topics on MSE, this subject haunts me.
My question is:
What do we need to define a category?
According to some books, a category consists of a class $\text{Obj}$ of objects and a set $\text{Hom}$ of morphisms which satisfy some axioms. For me it means, that to define a category we need some set theory. But there are many different set theories. Do they raise different category theories?
Also, as I understand, when we are talking about specific categories, like $\text{Set}$, $\text{Grp}$,... we mean models (interpretations) of the axioms of a category. Is it correct?