Combinatorial categorical equivalences of Dold-Kan type

We prove a class of equivalences of additive functor categories that are relevant to enumerative combinatorics, representation theory, and homotopy theory. Let 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 X are equivalent to chain complexes in X; (b) the observation of Church, Ellenberg and Farb [9] that X-valued species are equivalent to X-valued functors from the category of finite sets and injective partial functions; (c) a result T. Pirashvili calls of "Dold-Kan type" and so on. When X is semi-abelian, we prove the adjunction that was an equivalence is now at least monadic, in the spirit of a theorem of D. Bourne.