What is a projective object in $\rm Set$? What property of a set in $\sf{ZF}$ is equivalent to its being a projective object in the category $\rm Set$? Since all sets are projective assuming $\sf AC$ my guess is that it is equivalent to well-orderablility, but a direct transcription of the definition seems different from the definition of well-orderability, so I'm not certain.
If $X$ is a projective object in $\rm Set$, then this means that for every surjection $e:P\to Q$ and every function $f:X\to Q$ there is a function $h:X\to P$ such that $e\circ h=f$.

A set $Q$ is projective iff the axiom of choice is true for collections of $|Q|$ sets.  Indeed, if $e:P\to Q$ is a surjection, then the fibers of $e$ are a collection of $|Q|$ nonempty sets, and a choice function for this collection is the same as a right inverse to $e$.  Conversely, if $S$ is a collection of $|Q|$ nonempty sets and $f:S\to Q$ is a bijection, then let $P$ be the disjoint union of the elements of $S$ and let $e:P\to Q$ be the map induced by $f$.  Then $e$ is surjective, and a right inverse for $e$ gives a choice function for $S$.
So in particular, for instance, finite sets are always projective, and countably infinite sets are projective iff the countable axiom of choice is true.
