  Cartesian differential categories come equipped with a differential operator
which formalises the total derivative from multivariable calculus. Cofree
Cartesian differential categories always exist over a specified base category,
where the general construction is based on Faa di Bruno's formula. A natural
question to ask is, when given an arbitrary Cartesian differential category,
how can one check if it is cofree without knowing the base category? In this
paper, we provide characterisations of cofree Cartesian differential categories
without specifying a base category. The key to these characterisations is,
surprisingly, maps whose derivatives are zero, which we call differential
constants. One characterisation is in terms of the homsets being complete
ultrametric spaces, where the ultrametric is induced by differential constants,
which is similar to the metric for power series. Another characterisation is as
algebras of a monad. In either characterisation, the base category is the
category of differential constants. We also discuss other basic properties of
cofree Cartesian differential categories, such as the linear maps, and explain
how many well-known Cartesian differential categories (such as polynomial or
smooth functions) are not cofree.
