Is there a known way to built category theory on sound foundation, as occurs with axiomatic set theory ?

As far as I know, there exists at least to ways to talk about category theory :

1- to consider each Hom(X,Y) as an element of ZFC. Does this lead to some paradoxes ? 2- to avoid the use of ZFC. Can that be done in a rigourous manner ?

I know that there has been a debate about these foundation issues in the 60's and 70's among mathematicians (small categories vs big categories, use by Grothendieck of the notion of "universes" to link properly set theory to category theory, work of Benabou on the foundations of category theory...) , but is this debate closed today ?

Can you recommend some recent bibliography on that matter ?

Is there a known way to built category theory on sound foundation, as occurs with axiomatic set theory ?

As far as I know, there exists at least to ways to talk about category theory :

1. To consider each Hom(X,Y) as an element of ZFC. Does this lead to some paradoxes ?
2. To avoid the use of ZFC. Can that be done in a rigourous manner ?

I know that there has been a debate about these foundation issues in the 60's and 70's among mathematicians (small categories vs big categories, use by Grothendieck of the notion of "universes" to link properly set theory to category theory, work of Benabou on the foundations of category theory...) , but is this debate closed today ?

Can you recommend some recent bibliography on that matter ?