is it possible to characterize the elements of a (special) direct limit only using the universal property? in detail:

let's first concentrate on the category of sets. by an element, I mean a morphism defined on the terminal object $\{*\}$. let $A_1 \to A_2 \to ...$ be a sequence of sets, and $A$ their colimit which is defined by the universal property (morphisms on $A$ are compatible morphisms on the $A_n$). can we prove, **without using an explicit construction** of $A$, that every element of $A$ comes from an element in one of the $A_n$ and that two such elements are equal iff they are equal in some greater $A_m$?

for example, it can be shown without using an explicit construction that the coproduct $M+N$ of two sets $M,N$ (or an arbitrary family of sets) consists of the elements of $M$ and of $N$ and they exclude each other: denote $i_M,i_N$ the units ("inclusions"), then it is easy to see that $M,N \to im(i_M) \cup im(i_N)$ also satisfy the universal property so that $M + N = im(i_M) \cup im(i_N)$. now if $x \in im(i_M) \cap im(i_N)$, there are $m \in M, n \in N$ such that $x = i_M(m) = i_N(n)$. define $M \to \{0,1\}$ by sending $m$ to $0$, and the rest to $1$, and $N \to \{0,1\}$ by sending $n$ to $1$, and the rest to $0$. this yields $M + N \to \{0,1\}$ sending $x$ to $0$ and to $1$, contradiction.

perhaps this is not possible for directed limits because we have to use at least some specific properties of sets. but what about topoi or cosmoi (bicomplete closed symmetric monoidal categories)? I'm mainly interested in sets here, but perhaps more abstractions are needed to make clear what the goal is: a pure categorical approach to the usual constructions.

here is a functorial reformulation: let $... \to A_2 \to A_1$ be a sequence of (representable) endofunctors of sets and $A$ their explicitely constructed limit, i.e. $A(M)$ consists of compatible elements of the $A_n(M)$. how can we prove directly that every natural transformation $A \to id$ factors through one of the projections $A \to A_n$?

even if it is impossible, let's take this as a sort of axiom and consider algebraic structures (the usual ones, over sets). can we prove that the forgetful functor preserves direct limits of $A_1 \to A_2 \to ...$? again, without already knowing it by an explicit construction.

ps: there are lots of related questions. they are in the same spirit as my earlier one which has not been answered (although the rules for the bounty decided to accept an answer).

twobackslashes to write a set brace. – Mariano Suárez-Alvarez♦ Jan 6 '10 at 17:29isthe background theory you are using to prove these assertedly clear statements? – Todd Trimble♦ Jan 6 '10 at 18:29