Let's work in a set theory without assuming AC (for instance, but not necessarily, ZF). Fix a set $k$ satisfying $k\times k \simeq k$, and consider its powerset $X = 2^k$. I have a technical condition that is satisfied whenever $X$ is well-orderable, but I can't tell what other examples there might be. Excuse the peculiar phrasing: I'm aiming for the most constructive means possible (by "well-orderable" I mean classically so: a total order such that every inhabited subset has a least element. This is equivalent to $X$ being a [choice object](http://ncatlab.org/nlab/show/choice+object)). 


Let $p\colon X \to I$ be a partition of $X$ into subsets $X\_i=p^{-1}(i)\lt X$, $i\in I$ such that the images of all functions $f\_i\colon X\_i \to X$ have inhabited complement $\forall i \in I$. Assume I is minimal in the $\lt$-ordering of sets by cardinality (if $X$ is well-orderable, then $I$ is unique and $I\simeq cf(X)$). The condition is as follows:



> 
> For all partitions as just described, and all families of functions $(f\_i\colon X\_i \to X\mid i \in I)$, the surjection
>  $$
>  \coprod\_{i\in I} X\_i \setminus f\_i(X\_i) \to I
> $$
>  has a section.
> 
> 
> 


The consequence of this that I want is that I can choose a point in the complement of every map $X \to X^I$, namely $X < X^I$. For well-orderable $X$ this is a corollary of a restriction of König's theorem to powers of well-orderable sets in place of arbitrary products of families.


I would like to know if there are examples that aren't well-orderable, or otherwise how close to be able to well-order $X$ this gets. Note in particular, that we could take every function $f\_i$ to be constant at a fixed point $x\in X$, and so this case implies that we should be able to take a section of $X\setminus\lbrace x\rbrace \to I$ and hence of $p$, for $p$ and $I$ as above.


Note that I don't quite have the result that $X$ is a choice object (equivalently, classically well-orderable). I'm not demanding the existence of sections for arbitrary surjections, or more generally for total relation into $X$ (a fine distinction, I'm sure). 


**Edit:** $X$ being a choice object is equivalent to saying that there is a choice function from the set $P\_+X$ of inhabited subsets to $X$. I am only asking for a choice function on the subset of $P\_+X$ given by those subsets of $X$ of strictly smaller cardinality, call it $P^{\lt X}\_+X$.



How close to being well-orderable does this make my powerset?
Prompted by the discussion in comments above, Asaf wrote a blog post [Cofinality and the axiom of choice](http://karagila.org/2015/cofinality-and-the-axiom-of-choice/) that gave the following example of a model of ZF:



> 
> It is consistent that every non well-orderable set has cofinality 2. This was shown by Monro (“[Decomposable Cardinals](https://eudml.org/doc/214599)“, Fund. Math. vol. 80 (1973), no. 2, 101–104.), more specifically the real numbers can have cofinality 2.
> 
> 
> 


This destroys the inequality $\kappa < 2^{cf(\kappa)}$ resulting from König's theorem (I like to call this inequality *König's corollary*), for the notion of cofinality in the original question.


