Classic work of Keimel (1970), Pierce (1967) and Dauns-Hofmann (1966) concerns dualities between commutative rings (with plenty of idempotents) and bundles of simpler rings over Stone spaces. Recent work of Armstrong et al. (2021) as well as the speaker has sought to find natural noncommutative extensions, thus putting the theory of Steinberg algebras/rings on a solid algebraic foundation. Parallel to this, we have the classic duality of Stone (1936) between Boolean algebras and Stone spaces, which has also been extended to noncommutative structures by Lawson-Kudryavtseva (2017), namely to inverse/restriction semigroups and ample groupoids/categories. It turns out that this can likewise be further extended to a duality of "Steinberg semigroups" and bundles of categories over ample groupoids, from which the Steinberg ring results are just a short step away. Here we outline how to do this, even in a functorial way, thus turning our previous work (and that of... more