  The Fa\`a di Bruno construction, introduced by Cockett and Seely, constructs
a comonad $\mathsf{Fa{\grave{a}}}$ whose coalgebras are precisely Cartesian
differential categories. In other words, for a Cartesian left additive category
$\mathbb{X}$, $\mathsf{Fa{\grave{a}}}(\mathbb{X})$ is the cofree Cartesian
differential category over $\mathbb{X}$. Composition in these cofree Cartesian
differential categories is based on the Fa\`a di Bruno formula, and corresponds
to composition of differential forms. This composition, however, is somewhat
complex and difficult to work with. In this paper we provide an alternative
construction of cofree Cartesian differential categories inspired by tangent
categories. In particular, composition defined here is based on the fact that
the chain rule for Cartesian differential categories can be expressed using the
tangent functor, which simplifies the formulation of the higher order chain
rule.
