> The key to understanding the fixpoint theorem, for me anyway, was realizing that when it says FG=G, it does not mean that for every F there is a G such that by calling F with parameter G one can get back G. I mention this in the blog post, above. The equality operator in FG=G is the equivalence relation induced by the calculus rewriting relation: that is, it's the symmetric reflexive transitive closure of the rewriting step relation. When you look at the proof of the theorem, it actually works by constructing an expression G such that evaluating G produces, as an intermediate result, FG. There is no need for F to even be a function, and if F is a function it doesn't matter what, if anything, F would actually do when called, because the proof of the theorem doesn't involve calling F.
I think the name "fixpoint combinator" is kindof bad, it it was called e.g. the "recursion combinator" I think people would find it more intuitive.
[0] https://fexpr.blogspot.com/2013/07/bypassing-no-go-theorems....