After reading about a half of the book, I tried to integrate my theory of funcoids with their theory of convergences.
And I noticed, that if I define convergences induced by funcoids following “Convergence of funcoids” chapter in my book, then convergences induced by (reflexive) funcoids are pretopologies (a narrow subclass of convergences). So convergences appear to be not a special case of funcoids. Certainly it seems that in the other direction funcoids are not a special case of convergences.
So we have a (probably difficult) problem: Find a common generalization of funcoids and convergences!