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

Characterisations of purity in a locally finitely presented additive category: A short functorial proof