∀ (n : ℕ) {X Y : SimplexCategoryᵒᵖ} (f : X ⟶ Y),
  CategoryTheory.CategoryStruct.comp ((SSet.stdSimplex.obj (SimplexCategory.mk n)).map f)
      ((fun d =>
            (SSet.stdSimplex.objEquiv.trans
                { toFun := fun f => ⋯.functor,
                  invFun := fun f =>
                    SimplexCategory.Hom.mk
                      (ULift.orderIso.{u_1, 0}.toOrderEmbedding.toOrderHom.comp (CategoryTheory.Functor.toOrderHom f)),
                  left_inv := ⋯, right_inv := ⋯ }).toIso)
          Y).hom =
    CategoryTheory.CategoryStruct.comp
      ((fun d =>
            (SSet.stdSimplex.objEquiv.trans
                { toFun := fun f => ⋯.functor,
                  invFun := fun f =>
                    SimplexCategory.Hom.mk
                      (ULift.orderIso.{u_1, 0}.toOrderEmbedding.toOrderHom.comp (CategoryTheory.Functor.toOrderHom f)),
                  left_inv := ⋯, right_inv := ⋯ }).toIso)
          X).hom
      ((CategoryTheory.nerve (ULift.{u_1, 0} (Fin (n + 1)))).map f)