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We know that the model theory of modules leads to a way of obtainingdefinable categories of modules over a ring $R$ as the kernels of certainfunctors $(R\textbf{-Mod})^{\text{op}}\to\textbf{Ab}$ rather than of functors$R\textbf{-Mod}\to\textbf{Ab}$ which are given by a pp pair. This paper willgive various algebraic characterisations of these functors in the case that $R$is an artin algebra.  Suppose that $R$ is an artin algebra. An additive functor$G:(R\textbf{-Mod})^{\text{op}}\to\textbf{Ab}$ preserves inverse limits and$G|_{(R\textbf{-mod})^{\text{op}}}:(R\textbf{-mod})^{\text{op}}\to\textbf{Ab}$is finitely presented if and only if there is a sequence of naturaltransformations $(-,A)\to(-,B)\to G\to 0$ for some $A,B\in R\textbf{-mod}$which is exact when evaluated at any left $R$-module. Any additive functor$(R\textbf{-Mod})^{\text{op}}\to\textbf{Ab}$ with one of these equivalentproperties has a definable kernel, and every definable subcategory of$R\textbf{-Mod}$ can be obtained as the kernel of a family of such functors.  In the final section a generalised setting is introduced, so that our resultsapply to more categories than those of the form $R\textbf{-Mod}$ for an artinalgebra $R$. That is, our results are extended to those locally finitelypresented $K$-linear categories whose finitely presented objects form adualising $K$-variety, where $K$ is a commutative artinian ring.

Duality and contravariant functors in the representation theory of artin algebras