Functoriality of similarity monoids for fractals

Let $C$ be the category whose objects are fractals satisfying the conditions of Theorem, with suitable homeomorphisms as arrows. Let $D$ be the category whose objects are Rees monoids, with suitable homomorphisms as arrows. Functoriality conjecture. There is a functor from $C$ to $D$. This would extend the preceding result that isometries between the fractals induce isomorphisms between their associated similarity monoids; the appropriate classes of homeomorphisms and homomorphisms still need to be specified and the existence of the functor remains open.