Where can I find the proof that the axiom of choice is equivalent to "every set is a projective object in $\bf Set$"?

Question

I've read it on Wikipedia that in the category of Sets, the statement "every set is a projective object" is equivalent to the Axiom of Choice.

I'd like some references with this proof but I'm having trouble finding them. Could anyone help me?

Answer 1

This is due to Andreas Blass,

Blass, Andreas "Injectivity, Projectivity, and the Axiom of Choice." Transactions of the American Mathematical Society, Vol. 255, (Nov., 1979), pp. 31-59

Answer 2

Let $E_{\alpha}$ be a collection of nonempty, pairwise disjoint sets, let $\Lambda$ be the index set, let $\cup_{\alpha \in \Lambda} E_{\alpha}$ be denoted by $E$. Consider the diagram with maps $id: \Lambda \rightarrow \Lambda$ and $e: E: \rightarrow \Lambda$ where $e(x)=\alpha \mbox{ if } x \in E_{\alpha}$, note that this function is well defined as the sets $E_{\alpha}$ are pairwise disjoint. Now $\Lambda$ projective implies there is a map $c : \Lambda \rightarrow E$ such that the diagram commutes which is essentially the axiom of choice. And Axiom of choice will imply there is such a map. Thus finishing the proof.