An overview of the categorical description of umbral calculus is given by Nigel Ray, in 
<A HREF="https://web.archive.org/web/20050321183609/http://www.ma.man.ac.uk/~nige/ucuc.pdf">Universal Constructions in Umbral Calculus</A> (1996).

> The most flexible basis for the study of umbral calculus lies in the
> category ${\cal C}_R$ of coassociative coalgebras over $R$, together
> with the category ${\cal A}_R$ of dual algebras. Additional features
> such as gradings and Hopf algebra structures may naturally be present
> in certain circumstances. This viewpoint was conceived by <A HREF="https://sites.math.washington.edu/~billey/classes/Hopf.algebra/bulletins/joni.rota.1979.pdf">Joni and Rota</A> (1979) and developed by Nichols and Sweedler (1982).