TITLE: Union of a object (a set) in the Elementary Theory of the Category of Sets
QUESTION [5 upvotes]: 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.

REPLY [5 votes]: 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.