Now the difficult one, the ontology axiom, which serves to place lower bounds on what exists.
Its useful to define galactic categories and then write the axiom in terms of that definition.

This is rather more difficult to state for category theoretic foundations than for set theoretic so its best to make some
definitions first.
The things we to defined are:

- transitive categories
- functorial abstraction
- limits and co-limits of functors
- functor space categories

The ontology axiom asserts that every functor belongs to some category which is transitive and which is closed under limits
and co-limits, functorial abstraction and construction of functor space categories.
Functorial abstraction plays a role similar to the replacement axiom in ZFC (which subsumes separation when suitably stated),
closure under limits and co-limits that if the closure under unions and intersections, closure under functor space categories
plays the role of closure under formation of powersets.

It seems better here to use a function rather than a relation for functorial abstraction, but this prevents the empty category
from being defined this way and we might need to assert the existence of the empty category.
However, asserting the existence of limits and co-limits is stronger than closure under unions and ought to give us the empty
category.
Its not so specific as the set theoretic union axiom, and therefore may not be sufficient for present purposes.