∀ {m n p : SimplexCategory} (f : m ⟶ n) (g : n ⟶ p),
  Opposite.op
      (SimplexCategory.Hom.mk
        { toFun := SimplexCategory.II.map' (SimplexCategory.Hom.toOrderHom (CategoryTheory.CategoryStruct.comp f g)),
          monotone' := ⋯ }) =
    CategoryTheory.CategoryStruct.comp
      (Opposite.op
        (SimplexCategory.Hom.mk
          { toFun := SimplexCategory.II.map' (SimplexCategory.Hom.toOrderHom f), monotone' := ⋯ }))
      (Opposite.op
        (SimplexCategory.Hom.mk
          { toFun := SimplexCategory.II.map' (SimplexCategory.Hom.toOrderHom g), monotone' := ⋯ }))