Is this graph-structure theory a candidate for a natural foundation of Set and Category theory?

[Zuhair Abdul Ghafoor Al-Johar]

Dear Sirs,

This topic comes as a continuation to earlier posts of
mine to FOM on structure theory as a foundation of
mathematics.

The following is what I believe to be a natural
axiomatization about graph structure, perhaps more
natural than earlier versions. The mereological
background is replaced by a theory about a simple
kind of set membership that mirrors atomic part-hood
in Mereology; that is of sets of Quine atoms, so only
Quine atoms are permitted to be elements of sets, all
sets are nonempty, and extensionality holds over sets
as usual, and there is nearly naive comprehension
that defines any collection of atoms satisfying a
property as long as that property holds of at least one atom.

The next primitive is that of "direction" which relates
uniquely an atom for every two atoms in a particular
order. So it can mirror ordered pairs of atoms, and
since those are themselves atoms, so we can define
relations as sets of them. Other axioms states that all
atoms are either nodes or directional edges, and a
node is never an arrow (directional edge), these serve
to simplify those graphs.

Now graphs are defined as sets closed under
node-hood, that is all nodes connected by arrows in
them are in them too. 

The structure of a graph is a function that is
determined by node-edge isomorphism between
graphs, so any two graphs have the same structure if
and only if they are isomorphic. Additionally, a
structure of a graph is stipulated to be isomorphic to
that graph, and also different continuous structures if
they are of a smaller size than the universe of all
nodes, then those would always be separate [disjoint]
from each other!

The axioms that give power to this theory is about
the size of the universe of all nodes in comparison to
smaller graphs. A graph is "small" if and only it has
less many nodes than the universe. 

The axioms asserts:
1. the existence of a small infinite graph
2. the amount of substructures of a small structure is small
3. the union of a small amount of small graphs is itself small 

An axiom of choice over continuous separate graphs
is stipulated, whereby we can select a node from every
continuous graph in any union of continuous
graphs that are separate [disjoint] from each other. 

For details formal exposition, see: https://sites.google.com/site/zuhairaljohar/structure-theory-of-graphs

This axiomatic system seems to be fairly natural and
simple, and consistent relative to known foundational
systems of set theory and category theory. Moreover it
does interpret ZFC. The category of all sets of ZFC is
definable as the set of all structures of small extensional
mono-rooted trees with finitely long branches, plus all
arrows between those structures including the identity
arrows over their nodes. By *extensional* tree, it's
meant that no node of it can have two distinct
isomorphic maximal subtrees stemming from it. The set
membership relation of ZFC can be defined over those
structures as structures of maximal subtrees whose root
nodes are those connected to the root node of the
main tree directly through arrows. So, it does provide an
explication about sets and their membership in the
standard sense of ZFC. I believe also that the category of
all small categories can as well be defined here in
almost straightforward manner. So, this theory can
serve as a natural foundation of both Set and Category theory.

Can we regard such a theory as a Candidate for a
foundational theory of mathematics?

P.S. earlier versions of graph-structure theory:
[1] Structure Theory as an alternative foundation
https://cs.nyu.edu/pipermail/fom/2014-March/017902.html

[2] A simplification of structure theory
https://cs.nyu.edu/pipermail/fom/2015-December/019381.html