I’ve been thinking about polynomial functors for a few years now, and at the beginning of that time I was also thinking about Dirichlet polynomials. The category 
has objects as on the left, and the category 
has objects as on the right: 
I wrote a couple of short notes with David Jaz Myers about these things, e.g. that the category of Dirichlet polynomials forms a topos. Indeed, it is equivalent to the topos of functions 
. That is, both and have interpretations as categories of bundles in 
, but with different sorts of maps: maps are forwards on base and forwards on fibers, whereas maps are forwards on base and backwards on fibers. I’ll explain this more in the next section.
This is a companion discussion topic for the original entry at https://topos.site/blog/2023-07-21-nuclear-adjunction-poly-dir/