Union of a object (a set) in the Elementary Theory of the Category of Sets I see the Todd Trimble article "Elementary Theory of the Category of Sets" on catlab.
I ask: how make (in the categorical setting) the usual union of a set $\cup X=${$y |\exists x\in X: y\in x$}?
This object is (strongly) no natural (no amenable to a functor and natural transformation). 
I ask this because the "Union axiom" is one of the ZF theory of sets.  
 A: The elements of sets in the theory ETCS are not other sets, they are functions $\ast \to X$. In particular, elements to not themselves have elements, so you cannot discuss the union as you have described it. In essence, this is one of the main difference between ETCS (a structural set theory) and ZF(C) (a material set theory).
The translation from ETCS is not simplistic. One has to take certain well-founded trees where the nodes are ETCS-sets to recover ZF(C)-sets. Actually, since ETCS is really of the strength of bounded Zermelo set theory with Choice you get this instead (see http://ncatlab.org/nlab/show/pure+set). Such a tree is actually a diagram $\mathcal{T} \to Set$ where $Set$ is given by ETCS, and represents a membership tree.
Then given a rooted tree (with root $T_0$) representing a material set, the union is (roughly) the tree with root the disjoint union of sets at the next level up from the original root and the union of all the branches above that.
A: There are several nice answers already, but François's answer makes me think that the following point might also be relevant.
It's a known fact that the ETCS axioms do not imply the existence of a coproduct
$$
\mathbb{N} + P(\mathbb{N}) + P(P(\mathbb{N})) + \cdots
$$
where $P$ means power set.  (At least, they don't unless they're inconsistent.)  So if that's the kind of "union" you had in mind, it's not always provided by ETCS.
However, you can add the following axiom scheme to ETCS (and indeed, ETCS plus this axiom scheme is equivalent to ZFC).  I'll say it a bit informally; a precise statement is in Section 8 of

Colin McLarty, Exploring categorical structuralism, Philosophia Mathematica 12 (2004), 37-53.

So: suppose you have a set $I$ and a family $(X_i)_{i \in I}$ of sets specified by a first-order formula.  The axiom states that the coproduct $\sum_{i \in I} X_i$ exists.  How does it state this?  By saying that there are a set $X$ (to be thought of as $\sum_{i \in I} X_i$) and a map $p: X \to I$ (to be thought of as the evident projection) such that for each $i \in I$, the fibre $p^{-1}(i)$ is isomorphic to $X_i$.
One moral of the variation between our answers: "union" and "disjoint union" are more different concepts than they might at first appear.
A: After reading your other question, I think I understand what you're looking for.
In ZF, one of the key purposes of the Union Axiom is to prove that the universe of sets (viewed as a topos) is closed under internal coproducts, i.e., that one can always form the disjoint union of a set-indexed family of sets.
Any elementary topos $\mathcal{E}$ is closed under internal coproducts and internal products: for any morphism $f:A \to B$, the functor $f^*:\mathcal{E}/B \to \mathcal{E}/A$ has both a left adjoint $\sum_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal coproduct) and a right adjoint $\prod_f:\mathcal{E}/A\to\mathcal{E}/B$ (internal product). Note that internal coproducts are conceptually stronger than internal unions, though the existence proof for either in elementary toposes is essentially the same.
One can also formulate this in terms of indexed categories, where the topos $\mathcal{E}$ is used to index itself. This is closer to the usual notion of (set-indexed) coproduct, but working with slice categories is equivalent and technically simpler.
A: There is a sense in which there is a (trivial, tautological) axiom of (disjoint) union in the elementary theory of the category of sets. In any category with pullbacks, one can think of a slice above an object X (that is, a map into X) as representing an X-indexed set (the idea being that the set associated to any particular point in X is the fiber of that point under that map; pullback then acts as reindexing). One might then ask what the disjoint union of an indexed set is. Well, it will simply be the domain of the representing slice! (Since the domain of a map amounts to the same as the union of all its fibers).
(Of course, this does not touch upon the traditional idea of set theory as about "sets of sets of sets of sets...". To model such a theory within a theory of unstructured collections, you must use membership trees, as David notes.)
A: There are several articles that I wrote on ETCS, which had originally appeared on the (currently inactive) blog Topological Musings. The nLab articles are nothing more than transcriptions of what I had written into MathML, which is what we use at the nLab. They stop a little short of what you are asking for specifically, so perhaps I can fill the gap now, and say how I think I might have proceeded. 
As already mentioned by David and Sridhar, ETCS differs from traditional set theories that are based on a global membership relation (theories whose underlying signature consists of a single binary relation $\in$). Instead, ETCS spells out axioms that one expects to hold for a category of sets and functions. For those who speak the language, the axioms amount to saying that a model of ETCS is a topos with a natural numbers object, such that the terminal set is a generator and the axiom of choice ("epis split") holds. 
In this framework, one treats "union" as an operation which internalizes the external operation of taking joins in subset lattices. Thus, if $X$ is a set (or an object if you like), the union operation relative to $X$ is an appropriate morphism 
$$\bigcup: PPX \to PX$$ 
where $PX$ denotes the power set/object of $X$. By the universal property of power objects, this morphism corresponds to a subobject of $X \times PPX$. This subobject is specified by the formula (of an internal language for toposes) 
$$\exists_{A: PX} (x \in_X A) \wedge (A \in_{PX} C)$$ 
where $x$ is of type $X$ and $C$ is of type $PPX$. 
There are several ways of doing this, even if one is not familiar with the internal language of a topos. One way, which works for general toposes, proceeds by interpreting the quantifier $\exists_{A: PX}$ directly in terms of image factorizations. Namely, consider the image factorization of the composite 
$$[(x \in_X A) \wedge (A \in_{PX} C)] \hookrightarrow X \times PX \times PPX \stackrel{proj}{\to} X \times PPX$$ 
to get the desired subobject $I \hookrightarrow X \times PPX$. (Of course, this requires that one construct image factorizations in a topos, as treated in any standard text.) The subobject described in brackets is, in turn, a pullback of the form 
$$(1_X \times \delta \times 1_{PPX})^\ast(\in_X \times \in_{PX})$$ 
where $\in_X \hookrightarrow X \times PX$ and $\in_{PX} \hookrightarrow PX \times PPX$ are the canonical subobjects, and where $\delta: PX \to PX \times PX$ is the diagonal. Then, as said before, the map $PPX \to PX$ which classifies this image $I \hookrightarrow X \times PPX$ is the desired internal union relative to $X$. 
The second way to go is to realize that a model of ETCS is in particular a Boolean topos. Then, if one has already constructed universal quantification (see for instance the second of the three articles in the ETCS series), one can easily interpret the formula 
$$\neg \forall_{A: PX} (x \in_X A) \Rightarrow \neg(A \in_{PX} C)$$ 
once one has defined internal negation, which is not difficult. This circumvents the need to first construct images, but only works in the Boolean case. 
However one spells out the details, the larger point is that in ETCS, membership relations are local and relative to objects $X$, in the form of universal subobjects $\in_X \hookrightarrow X \times PX$, as opposed to being given by a single global relation $\in$ that obtains on the class of objects. Correspondingly, set-theoretic operations like union and intersection are also local and relative in this sense. Otherwise, the first-order formulas that specify such operations -- the ones we all know and love -- work pretty much the same way; in ETCS, the relevant operations may be constructed by clever exploitation of universal properties of relations $\in_X$, and not just asserted to exist by way of a comprehension or separation axiom scheme. 
