Combinatorial categorical equivalences of Dold-Kan type

In this paper we prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let $\mathcal{X}$ denote an additive category with finite direct sums and splitting idempotents. The class includes (a) the Dold-Puppe-Kan theorem that simplicial objects in $\mathcal{X}$ are equivalent to chain complexes in $\mathcal{X}$; (b) the observation of Church-Ellenberg-Farb that $\mathcal{X}$-valued species are equivalent to $\mathcal{X}$-valued functors from the category of finite sets and injective partial functions; (c) a Dold-Kan-type result of Pirashvili concerning Segal's category $\Gamma$; and so on. We provide a construction which produces further examples.