  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 $Poly_E$.
