I’ve found a counterexample to the following conjecture:

Statement For every composable funcoids $f$ and $g$ we have

$H \in \mathrm{up}(g \circ f) \Rightarrow \exists F \in \mathrm{up}\, f, G \in \mathrm{up}\, g : H \in\mathrm{up}\, (G \circ F) .$

The counterexample is $f=a\times^{\mathsf{FCD}} \{p\}$ and $g=\{p\}\times^{\mathsf{FCD}}a$, $H=1$ where $a$ is an arbitrary nontrivial ultrafilter and $p$ is an arbitrary point.

I leave the proof that it is a counterexample as an easy exercise for the reader (however I am going to add the proof to my book soon).

That the conjecture failed invalidates my new proof of Urysohn’s lemma which was based on this conjecture.