  In this paper, we consider abelian functor calculus, the calculus of functors
of abelian categories established by the second author and McCarthy. We
carefully construct a category of abelian categories and suitably homotopically
defined functors, and show that this category, equipped with the directional
derivative, is a cartesian differential category in the sense of Blute,
Cockett, and Seely. This provides an abstract framework that makes certain
analogies between classical and functor calculus explicit. Inspired by Huang,
Marcantognini, and Young's chain rule for higher order directional derivatives
of functions, we define a higher order directional derivative for functors of
abelian categories. We show that our higher order directional derivative is
related to the iterated partial directional derivatives of the second author
and McCarthy by a Fa\`a di Bruno style formula. We obtain a higher order chain
rule for our directional derivatives using a feature of the cartesian
differential category structure, and with this provide a formulation for the
$n$th layers of the Taylor tower of a composition of functors $F\circ G$ in
terms of the derivatives and directional derivatives of $F$ and $G$,
reminiscent of similar formulations for functors of spaces or spectra by Arone
and Ching. Throughout, we provide explicit chain homotopy equivalences that
tighten previously established quasi-isomorphisms for properties of abelian
functor calculus.
