∀ {C : Type u_1} [inst : CategoryTheory.Category.{v_1, u_1} C] (X : CategoryTheory.SimplicialObject.Augmented C)
  {X_1 Y : (CategoryTheory.WithInitial SimplexCategory)ᵒᵖ} (f : X_1 ⟶ Y),
  (AugmentedSimplexCategory.equivAugmentedSimplicialObject.inverse.obj X).map f =
    match
      match Opposite.unop X_1 with
      | CategoryTheory.WithInitial.of x => CategoryTheory.WithTerminal.of (Opposite.op x)
      | CategoryTheory.WithInitial.star => CategoryTheory.WithTerminal.star,
      match Opposite.unop Y with
      | CategoryTheory.WithInitial.of x => CategoryTheory.WithTerminal.of (Opposite.op x)
      | CategoryTheory.WithInitial.star => CategoryTheory.WithTerminal.star,
      match X_1, Y, Opposite.unop f, f with
      | Opposite.op (CategoryTheory.WithInitial.of x), Opposite.op (CategoryTheory.WithInitial.of y), f, x_1 =>
        (CategoryTheory.WithTerminal.down f).op
      | Opposite.op (CategoryTheory.WithInitial.of a), Opposite.op CategoryTheory.WithInitial.star, x, x_1 =>
        CategoryTheory.WithTerminal.starTerminal.from (CategoryTheory.WithTerminal.of (Opposite.op a))
      | Opposite.op CategoryTheory.WithInitial.star, Opposite.op CategoryTheory.WithInitial.star, x, x_1 =>
        CategoryTheory.CategoryStruct.id CategoryTheory.WithTerminal.star with
    | CategoryTheory.WithTerminal.of a, CategoryTheory.WithTerminal.of a_1, f =>
      X.left.map (CategoryTheory.WithTerminal.down f)
    | CategoryTheory.WithTerminal.of x, CategoryTheory.WithTerminal.star, x_1 => X.hom.app x
    | CategoryTheory.WithTerminal.star, CategoryTheory.WithTerminal.star, x => CategoryTheory.CategoryStruct.id X.right