A codifferential category is a symmetric monoidal k-linear category endowed with a special kind of monad which allows it to interpret differential calculus. Part of this structure is that of an "algebra modality"—this means that it is a monad with a map from the free commutative monoid monad. We explain how the notion of codifferential category arises automatically from two more basic ingredients: the 2-category of algebra modalities and lax morphisms; and the process of adjoining a nilsquare object to a symmetric monoidal category.