So this is what I learned from all that.
A function is a special kind of relation. So, when you get a feasibility relation from a monotone function \\(F: X \to Y\\) by taking its companion, you might be fooled into thinking this relation is just that function thought of as a relation in the usual way. But it's not: the usual way gives the relation \\(f(x) = y \\), but here we're getting the relation \\(f(x) \le y\\).
That should have been obvious - but my mental image of which feasibility relations arise as companions was a bit off.
I think one of the feasibility relations in my puzzles is neither a companion nor a conjoint. Maybe I'm confused. But does someone see which one I mean?