>>11199237>>11199301>are there any category theory axioms in the same way sets are rigorously defined?Yes. I've some notes on here here, especially at the beginning for FOL, and see the links therein at the bottom (mostly the nLab ones)
https://axiomsofchoice.org/category_theory>>11199119I'm not sure if you want to consider isomorphic graphs or whether you mean to consider all the graphs for different categories at once.
Once this is given proper form, it's mostly just a nasty question of size and will thus on your setp.
To help making it more concrete (pun intended), note that the number of edges in a graph will be some cardinal and the consider the realization of a graph is a functor:
Consider the (abstract) category C with two objects (call them e and v) and four arrows, two of which are parallel
id_e : e->e
id_v : v->v,
s : e->v
t: e->v
Consider now a set valued functor
F : C ---> Sets
which is a map taking the objects e and v to particular sets (say two cardinals Fe:=E and Fv:=V in Sets) and s and t to two functions (say F(s):=S and F(t):=T, of type E->V).
Note that this exactly is the specification of a directed graph with |V| vertices V and |E| edges E with fixed source and target specification (given by the functions S and T).
All the grpahs are just all those functors. Since those are just the presheaves on the tiny C^op, the collection of F's together as a category is actually a nice topos.
WIth that we're closer to getting a feel for it's size (Sets to the power of something a bit more complicated than the discrete two objects category) w.r.t. whatever "Sets" meant for us.
There's a paper by Schulman on size issues, maybe it helps:
https://arxiv.org/pdf/0810.1279.pdf 
