AlgebraicTopology.DoldKan.identity_N₂_objectwise

theorem AlgebraicTopology.DoldKan.identity_N₂_objectwise : ∀ {C : Type u_1} [inst : CategoryTheory.Category.{u_2, u_1} C] [inst_1 : CategoryTheory.Preadditive C] [inst_2 : CategoryTheory.Limits.HasFiniteCoproducts C] (P : CategoryTheory.Idempotents.Karoubi (CategoryTheory.SimplicialObject C)), CategoryTheory.CategoryStruct.comp (AlgebraicTopology.DoldKan.N₂Γ₂.inv.app (AlgebraicTopology.DoldKan.N₂.obj P)) (AlgebraicTopology.DoldKan.N₂.map (AlgebraicTopology.DoldKan.Γ₂N₂.natTrans.app P)) = CategoryTheory.CategoryStruct.id (AlgebraicTopology.DoldKan.N₂.obj P)

This theorem, named `AlgebraicTopology.DoldKan.identity_N₂_objectwise`, asserts that for any category `C` which is preadditive and has finite coproducts, and for any object `P` in the Karoubi envelope of the category of simplicial objects in `C`, the composition of the inverse of the application of `N₂Γ₂` to `N₂.obj P` and the map of `N₂` applied to `Γ₂N₂.natTrans.app P` is the identity on `N₂.obj P`. In simpler terms, this theorem is about the interaction of certain functors and natural transformations within the area of algebraic topology known as Dold-Kan correspondence.

More concisely: For any preadditive category `C` with finite coproducts and any object `P` in the Karoubi envelope of the category of simplicial objects in `C`, the composition of `N₂.obj (Γ₂N₂.natTrans.app P)` and the inverse of `N₂Γ₂ (N₂.obj P)` is the identity on `N₂.obj P` in the Dold-Kan correspondence.