  In this expository article, we will give an efficient functorial proof of the
equivalence of various characterisations of purity in a finitely accessible
additive category $\mathcal C$. The complications of the proofs for specific
choices of $\mathcal C$ are contained in the description of fp-injective and
injective objects in $(\operatorname{fp}\mathcal C,\mathrm{Ab})$, the category
of additive functors $\mathcal C\to\mathrm{Ab}$. For example, the equivalence
of many characterisations of purity in a module category $\mathcal
A\mathrm{-Mod}$ is a simple corollary of what we will prove here, since we know
which objects are fp-injective, and which objects are injective, in $(\mathcal
A\mathrm{-mod},\mathrm{Ab})$.
