There are some existential axioms in set theory, for example, axiom schema of specification.
It's my understanding that category theory isn't based essentially on set theoretic foundation.
If so, I think there must be some existential axioms for category theory but I couldn't find description anywhere.
Do you know works about this subject?
The question is a bit vague, but you might be interested in the following: There is a structural, category-theoretical foundational system for mathematics, namely Elementary Theory of the Category of Sets. As you have speculated, it is indeed the case that in this formal system you have several category-theoretical existential axioms, for example the existence of pull-backs or the existence of a natural-number object. The existence of pullbacks of monomorphisms is an analogue of the separation axiom in set-theory.