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Suppose that $\mathcal{E}$ is a well-pointed elementary topos, that $X$ and $Y$ are objects of $\mathcal{E}$, and that $F$ is a function which maps global elements $p: 1 \to X$ to global elements $F(p): 1 \to Y$ (here $1$ is the terminal object of $\mathcal{E}$). Does there exist a (necessarily unique) arrow $f: X \to Y$ in $\mathcal{E}$ such that $fp = F(p)$ for all $p$? Equivalently, is any object in a well-pointed topos the coproduct over its global elements of $1$? It's easy to show that the answer is "yes" if the coproduct exists since the induced map $\coprod_{p \in \Gamma X} 1 \to X$ is iso. But I don't know whether the coproduct exists in general.

(Could somebody with enough reputation create a "topos-theory" tag and add it to this? Thanks)

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up vote 1 down vote accepted

Perhaps consider a full subcategory of Set with just the functions you are allowed to create given ZF. Then find some function that in normal set theory requires AC. This should do the trick. Or use the topos of constructible sets and functions, same thing.

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Thanks. Someone on MathOverflow suggested a countable model of ZFC, which also works. – Rotwang Nov 20 '10 at 16:47

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