Structures on Categories of Polynomials

We define the monoidal category $(Poly_E,y,\triangleleft)$ of polynomials under composition in any category $E$ with finite limits, including both cartesian and vertical morphisms of polynomials, and generalize to this setting the Dirichlet tensor product of polynomials $\otimes$, duoidality of $\otimes$ and $\triangleleft$, closure of $\otimes$, and coclosures of $\triangleleft$. We also prove that $\triangleleft$-comonoids in $Poly_E$ are precisely the internal categories in $E$ whose source morphism is exponentiable, generalizing a result of Ahman-Uustalu equating categories with polynomial comonads, and show that coalgebras in this setting correspond to internal copresheaves. Finally, the double category of ``typed'' polynomials in $E$ is recovered using $\triangleleft$-bicomodules in $Poly_E$.