From the description of Category Theory in nlab:
Category theory is a structural approach to mathematics that can (through such methods as Lawvere's ETCS) provide foundations of mathematics and (through algebraic set theory) reproduce all the different axiomatic set theories; it does not need the concept of set to be formulated. Set theory is an analytic approach (element-wise) and can reproduce category theory by simply defining all the concepts in the usual way, as long as one include a technique to handle large categories (for instance by using classes instead of sets, or by including as an axiom that an uncountable inaccessible cardinal exists or even that Grothendieck universes exist).
That is the structural approach includes the analytic, and the analytic includes the structural. A little reflection shows that each theory includes images of itself and the other infinitely - somewhat like the mandelbrot set reproduces itself internally.
Given there are now two approaches to foundations, is it arguable that the 'true' Set Theory is something that is only represented in the two approaches? In the same way say that the number '9' is represented as nine or 1001?
Or is this an indication that there are in fact more than one Set Theory in the same way that denying the parallel postulate in non-euclidean geometry resulted in several different geometries: elliptic and spherical with the euclidean geometry occupying a special place because it is flat.
Certainly there is a similar picture in Topos Theory as a categorification of Set Theory (this is different to what is discussed above) where there are many Toposes but again the category of Sets occupy a unique place (I forget the characterisation of its uniqueness).