Can we capture all domains of discouse in the predicate logic within categorical logic?

Question

In the construction of the bounded quantifiers via adjoints in the fibered category of subsets over a set (see e.g. here on Wikipedia), is there any restriction on the sets - specifically regarding cardinality?

I essentially ask this because it feels like the aim is to fully describe first order predicate logic internally in a category - however as set theory is generally build over predicate logic, I'm not sure how natural it is to assume the logical universe of discourse to be some set from the start.

Answer 1

for this part of the question

the aim is to fully describe first order predicate logic internally in a category

there are papers very close to that theme at Steve Awodey's web site, including:

http://www.andrew.cmu.edu/user/awodey/preprints/fold.pdf
"In the present work, we generalize [Stone duality for Boolean algebras] from propositional to first-order logic. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms."

and

http://www.andrew.cmu.edu/user/awodey/preprints/BSL.pdf
"We show how to interpret the language of first-order set theory in an elementary topos endowed with, as extra structure, a directed structural system of inclusions (dssi).... we thus obtain a first-order set theory whose associated categories of sets are exactly the elementay toposes."