Inspired by this [answer][1] given by Noam, which (I think) implies that a set $A \in NP$ if and only if there is polynomial-time computable function $f$ from random string to an element in $A$ such that $A$= $\{ a: f(x)=a, x \in S \}$ where $S$ is the set of all witnesses $x$ for set $A$. If P=NP then every coNP set is _efficiently samplable_ in Noam's sense.


The following characterization of NP is taken from [Theory of computational complexity][2];

"A binary relation $R \subseteq\Sigma^* \times \Sigma^*$ is called polynomial honest if there exists a polynomial function $p$ such that $<x,y> \in R$ only if $|x|\le p(|y|)$ and $|y|\le p(|x|)$. A function $f:\Sigma^* \to \Sigma^*$ is polynomially honest  if the relation $\{ <x, f(x)>: x\in \Sigma^*\}$ is polynomial honest. 

Therefore, $A \subseteq \Sigma^*$ is in $NP$ if and only if $A=Range(f)$ for some polynomial honest and polynomial time computable function $f$."

Now, assume that there exists a _universal witness set_ $W \in P$ such that for every $NP$-complete set $D_i \subseteq \Sigma^*$ there is some polynomially honest and polynomial-time computable function $g_i:W\to \Sigma^*$ such that $D_i=Range(g_i)$.

> Does the existence of a _universal witness set_ for $NP$ implies the impossibility of _efficient sampling_ of coNP sets?
 



[1]: http://cstheory.stackexchange.com/a/567/495
[2]: https://books.google.com.sa/books?id=oG6lRcwRqCUC&pg=PA71&dq=%22polynomially+honest%22++%22honest+function%22&hl=en&sa=X&ved=0CCkQ6AEwAmoVChMI9snDzO7kxwIVxH4aCh0A8AjC#v=onepage&q=%22polynomially%20honest%22%20%20%22honest%20function%22&f=false