∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] [inst_1 : CategoryTheory.Preadditive C]
  [inst_2 : CategoryTheory.IsIdempotentComplete C] [inst_3 : CategoryTheory.Limits.HasFiniteCoproducts C]
  {X Y : CategoryTheory.SimplicialObject C} (f : X ⟶ Y),
  CategoryTheory.Idempotents.DoldKan.N.map f =
    (CategoryTheory.Idempotents.toKaroubiEquivalence (ChainComplex C ℕ)).inverse.map
      (AlgebraicTopology.DoldKan.N₁.map f)