ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- CategoryTheory.op_hom_leftUnitor
@[simp] lemma op_hom_leftUnitor (X : C) : (λ_ X).hom.op = (λ_ (op X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).hom.op = (CategoryTheory.MonoidalCategory.leftUnitor (Opposite.op X)).inv
-- CategoryTheory.unop_hom_leftUnitor
@[simp] lemma unop_hom_leftUnitor (X : Cᵒᵖ) : (λ_ X).hom.unop = (λ_ (unop X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).hom.unop =
  (CategoryTheory.MonoidalCategory.leftUnitor (Opposite.unop X)).inv
-- CategoryTheory.op_inv_leftUnitor
@[simp] lemma op_inv_leftUnitor (X : C) : (λ_ X).inv.op = (λ_ (op X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).inv.op = (CategoryTheory.MonoidalCategory.leftUnitor (Opposite.op X)).hom
-- CategoryTheory.unop_inv_leftUnitor
@[simp] lemma unop_inv_leftUnitor (X : Cᵒᵖ) : (λ_ X).inv.unop = (λ_ (unop X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).inv.unop =
  (CategoryTheory.MonoidalCategory.leftUnitor (Opposite.unop X)).hom
-- CategoryTheory.op_rightUnitor
@[simp] lemma op_rightUnitor (X : C) : (ρ_ X).op = (ρ_ (op X)).symm 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).op = (CategoryTheory.MonoidalCategory.rightUnitor (Opposite.op X)).symm
-- CategoryTheory.unop_rightUnitor
@[simp] lemma unop_rightUnitor (X : Cᵒᵖ) : (ρ_ X).unop = (ρ_ (unop X)).symm 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).unop =
  (CategoryTheory.MonoidalCategory.rightUnitor (Opposite.unop X)).symm
-- CategoryTheory.op_hom_rightUnitor
@[simp] lemma op_hom_rightUnitor (X : C) : (ρ_ X).hom.op = (ρ_ (op X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).hom.op =
  (CategoryTheory.MonoidalCategory.rightUnitor (Opposite.op X)).inv
-- CategoryTheory.unop_hom_rightUnitor
@[simp] lemma unop_hom_rightUnitor (X : Cᵒᵖ) : (ρ_ X).hom.unop = (ρ_ (unop X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).hom.unop =
  (CategoryTheory.MonoidalCategory.rightUnitor (Opposite.unop X)).inv
-- CategoryTheory.op_inv_rightUnitor
@[simp] lemma op_inv_rightUnitor (X : C) : (ρ_ X).inv.op = (ρ_ (op X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).inv.op =
  (CategoryTheory.MonoidalCategory.rightUnitor (Opposite.op X)).hom
-- CategoryTheory.unop_inv_rightUnitor
@[simp] lemma unop_inv_rightUnitor (X : Cᵒᵖ) : (ρ_ X).inv.unop = (ρ_ (unop X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).inv.unop =
  (CategoryTheory.MonoidalCategory.rightUnitor (Opposite.unop X)).hom
-- CategoryTheory.op_tensor_op
theorem op_tensor_op {W X Y Z : C} (f : W ⟶ X) (g : Y ⟶ Z) : f.op ⊗ g.op = (f ⊗ g).op 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {W X Y Z : C} (f : W ⟶ X) (g : Y ⟶ Z)
Goal: CategoryTheory.MonoidalCategory.tensorHom f.op g.op = (CategoryTheory.MonoidalCategory.tensorHom f g).op
-- CategoryTheory.unop_tensor_unop
theorem unop_tensor_unop {W X Y Z : Cᵒᵖ} (f : W ⟶ X) (g : Y ⟶ Z) :
    f.unop ⊗ g.unop = (f ⊗ g).unop 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {W X Y Z : Cᵒᵖ} (f : W ⟶ X) (g : Y ⟶ Z)
Goal: CategoryTheory.MonoidalCategory.tensorHom f.unop g.unop = (CategoryTheory.MonoidalCategory.tensorHom f g).unop
-- CategoryTheory.mop_tensorObj
@[simp] lemma mop_tensorObj (X Y : C) : mop (X ⊗ Y) = mop Y ⊗ mop X 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y : C)
Goal: { unmop := CategoryTheory.MonoidalCategory.tensorObj X Y } =
  CategoryTheory.MonoidalCategory.tensorObj { unmop := Y } { unmop := X }
-- CategoryTheory.unmop_tensorObj
@[simp] lemma unmop_tensorObj (X Y : Cᴹᵒᵖ) : unmop (X ⊗ Y) = unmop Y ⊗ unmop X 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.tensorObj X Y).unmop = CategoryTheory.MonoidalCategory.tensorObj Y.unmop X.unmop
-- CategoryTheory.mop_tensorUnit
@[simp] lemma mop_tensorUnit : mop (𝟙_ C) = 𝟙_ Cᴹᵒᵖ 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C]
Goal: { unmop := 𝟙_ C } = 𝟙_ Cᴹᵒᵖ
-- CategoryTheory.unmop_tensorUnit
@[simp] lemma unmop_tensorUnit : unmop (𝟙_ Cᴹᵒᵖ) = 𝟙_ C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C]
Goal: (𝟙_ Cᴹᵒᵖ).unmop = 𝟙_ C
-- CategoryTheory.mop_tensorHom
@[simp] lemma mop_tensorHom {X₁ Y₁ X₂ Y₂ : C} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂) :
    (f ⊗ g).mop = g.mop ⊗ f.mop 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X₁ Y₁ X₂ Y₂ : C} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂)
Goal: (CategoryTheory.MonoidalCategory.tensorHom f g).mop = CategoryTheory.MonoidalCategory.tensorHom g.mop f.mop
-- CategoryTheory.unmop_tensorHom
@[simp] lemma unmop_tensorHom {X₁ Y₁ X₂ Y₂ : Cᴹᵒᵖ} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂) :
    (f ⊗ g).unmop = g.unmop ⊗ f.unmop 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X₁ Y₁ X₂ Y₂ : Cᴹᵒᵖ} (f : X₁ ⟶ Y₁) (g : X₂ ⟶ Y₂)
Goal: (CategoryTheory.MonoidalCategory.tensorHom f g).unmop = CategoryTheory.MonoidalCategory.tensorHom g.unmop f.unmop
-- CategoryTheory.mop_whiskerLeft
@[simp] lemma mop_whiskerLeft (X : C) {Y Z : C} (f : Y ⟶ Z) :
    (X ◁ f).mop = f.mop ▷ mop X 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C) {Y Z : C} (f : Y ⟶ Z)
Goal: (CategoryTheory.MonoidalCategory.whiskerLeft X f).mop =
  CategoryTheory.MonoidalCategory.whiskerRight f.mop { unmop := X }
-- CategoryTheory.unmop_whiskerLeft
@[simp] lemma unmop_whiskerLeft (X : Cᴹᵒᵖ) {Y Z : Cᴹᵒᵖ} (f : Y ⟶ Z) :
    (X ◁ f).unmop = f.unmop ▷ unmop X 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ) {Y Z : Cᴹᵒᵖ} (f : Y ⟶ Z)
Goal: (CategoryTheory.MonoidalCategory.whiskerLeft X f).unmop = CategoryTheory.MonoidalCategory.whiskerRight f.unmop X.unmop
-- CategoryTheory.mop_whiskerRight
@[simp] lemma mop_whiskerRight {X Y : C} (f : X ⟶ Y) (Z : C) :
    (f ▷ Z).mop = mop Z ◁ f.mop 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X Y : C} (f : X ⟶ Y) (Z : C)
Goal: (CategoryTheory.MonoidalCategory.whiskerRight f Z).mop =
  CategoryTheory.MonoidalCategory.whiskerLeft { unmop := Z } f.mop
-- CategoryTheory.unmop_whiskerRight
@[simp] lemma unmop_whiskerRight {X Y : Cᴹᵒᵖ} (f : X ⟶ Y) (Z : Cᴹᵒᵖ) :
    (f ▷ Z).unmop = unmop Z ◁ f.unmop 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] {X Y : Cᴹᵒᵖ} (f : X ⟶ Y) (Z : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.whiskerRight f Z).unmop = CategoryTheory.MonoidalCategory.whiskerLeft Z.unmop f.unmop
-- CategoryTheory.mop_associator
@[simp] lemma mop_associator (X Y Z : C) :
    (α_ X Y Z).mop = (α_ (mop Z) (mop Y) (mop X)).symm 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y Z : C)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).mop =
  (CategoryTheory.MonoidalCategory.associator { unmop := Z } { unmop := Y } { unmop := X }).symm
-- CategoryTheory.unmop_associator
@[simp] lemma unmop_associator (X Y Z : Cᴹᵒᵖ) :
    (α_ X Y Z).unmop = (α_ (unmop Z) (unmop Y) (unmop X)).symm 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y Z : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).unmop =
  (CategoryTheory.MonoidalCategory.associator Z.unmop Y.unmop X.unmop).symm
-- CategoryTheory.mop_hom_associator
@[simp] lemma mop_hom_associator (X Y Z : C) :
    (α_ X Y Z).hom.mop = (α_ (mop Z) (mop Y) (mop X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y Z : C)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).hom.mop =
  (CategoryTheory.MonoidalCategory.associator { unmop := Z } { unmop := Y } { unmop := X }).inv
-- CategoryTheory.unmop_hom_associator
@[simp] lemma unmop_hom_associator (X Y Z : Cᴹᵒᵖ) :
    (α_ X Y Z).hom.unmop = (α_ (unmop Z) (unmop Y) (unmop X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y Z : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).hom.unmop =
  (CategoryTheory.MonoidalCategory.associator Z.unmop Y.unmop X.unmop).inv
-- CategoryTheory.mop_inv_associator
@[simp] lemma mop_inv_associator (X Y Z : C) :
    (α_ X Y Z).inv.mop = (α_ (mop Z) (mop Y) (mop X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y Z : C)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).inv.mop =
  (CategoryTheory.MonoidalCategory.associator { unmop := Z } { unmop := Y } { unmop := X }).hom
-- CategoryTheory.unmop_inv_associator
@[simp] lemma unmop_inv_associator (X Y Z : Cᴹᵒᵖ) :
    (α_ X Y Z).inv.unmop = (α_ (unmop Z) (unmop Y) (unmop X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X Y Z : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.associator X Y Z).inv.unmop =
  (CategoryTheory.MonoidalCategory.associator Z.unmop Y.unmop X.unmop).hom
-- CategoryTheory.mop_leftUnitor
@[simp] lemma mop_leftUnitor (X : C) : (λ_ X).mop = (ρ_ (mop X)) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).mop = CategoryTheory.MonoidalCategory.rightUnitor { unmop := X }
-- CategoryTheory.unmop_leftUnitor
@[simp] lemma unmop_leftUnitor (X : Cᴹᵒᵖ) : (λ_ X).unmop = ρ_ (unmop X) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).unmop = CategoryTheory.MonoidalCategory.rightUnitor X.unmop
-- CategoryTheory.mop_hom_leftUnitor
@[simp] lemma mop_hom_leftUnitor (X : C) : (λ_ X).hom.mop = (ρ_ (mop X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).hom.mop =
  (CategoryTheory.MonoidalCategory.rightUnitor { unmop := X }).hom
-- CategoryTheory.unmop_hom_leftUnitor
@[simp] lemma unmop_hom_leftUnitor (X : Cᴹᵒᵖ) : (λ_ X).hom.unmop = (ρ_ (unmop X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).hom.unmop = (CategoryTheory.MonoidalCategory.rightUnitor X.unmop).hom
-- CategoryTheory.mop_inv_leftUnitor
@[simp] lemma mop_inv_leftUnitor (X : C) : (λ_ X).inv.mop = (ρ_ (mop X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).inv.mop =
  (CategoryTheory.MonoidalCategory.rightUnitor { unmop := X }).inv
-- CategoryTheory.unmop_inv_leftUnitor
@[simp] lemma unmop_inv_leftUnitor (X : Cᴹᵒᵖ) : (λ_ X).inv.unmop = (ρ_ (unmop X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.leftUnitor X).inv.unmop = (CategoryTheory.MonoidalCategory.rightUnitor X.unmop).inv
-- CategoryTheory.mop_rightUnitor
@[simp] lemma mop_rightUnitor (X : C) : (ρ_ X).mop = (λ_ (mop X)) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).mop = CategoryTheory.MonoidalCategory.leftUnitor { unmop := X }
-- CategoryTheory.unmop_rightUnitor
@[simp] lemma unmop_rightUnitor (X : Cᴹᵒᵖ) : (ρ_ X).unmop = λ_ (unmop X) 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).unmop = CategoryTheory.MonoidalCategory.leftUnitor X.unmop
-- CategoryTheory.mop_hom_rightUnitor
@[simp] lemma mop_hom_rightUnitor (X : C) : (ρ_ X).hom.mop = (λ_ (mop X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).hom.mop =
  (CategoryTheory.MonoidalCategory.leftUnitor { unmop := X }).hom
-- CategoryTheory.unmop_hom_rightUnitor
@[simp] lemma unmop_hom_rightUnitor (X : Cᴹᵒᵖ) : (ρ_ X).hom.unmop = (λ_ (unmop X)).hom 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).hom.unmop = (CategoryTheory.MonoidalCategory.leftUnitor X.unmop).hom
-- CategoryTheory.mop_inv_rightUnitor
@[simp] lemma mop_inv_rightUnitor (X : C) : (ρ_ X).inv.mop = (λ_ (mop X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).inv.mop =
  (CategoryTheory.MonoidalCategory.leftUnitor { unmop := X }).inv
-- CategoryTheory.unmop_inv_rightUnitor
@[simp] lemma unmop_inv_rightUnitor (X : Cᴹᵒᵖ) : (ρ_ X).inv.unmop = (λ_ (unmop X)).inv 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalCategory.rightUnitor X).inv.unmop = (CategoryTheory.MonoidalCategory.leftUnitor X.unmop).inv
-- CategoryTheory.MonoidalOpposite.mopEquiv_inverse
@[simps] def MonoidalOpposite.mopEquiv : C ≌ Cᴹᵒᵖ 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C]
Goal: (CategoryTheory.MonoidalOpposite.mopEquiv C).inverse = CategoryTheory.unmopFunctor C
-- CategoryTheory.MonoidalOpposite.mopEquiv_counitIso
@[simps] def MonoidalOpposite.mopEquiv : C ≌ Cᴹᵒᵖ 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C]
Goal: (CategoryTheory.MonoidalOpposite.mopEquiv C).counitIso =
  CategoryTheory.Iso.refl (CategoryTheory.unmopFunctor C ⋙ CategoryTheory.mopFunctor C)
-- CategoryTheory.MonoidalOpposite.mopEquiv_functor
@[simps] def MonoidalOpposite.mopEquiv : C ≌ Cᴹᵒᵖ 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C]
Goal: (CategoryTheory.MonoidalOpposite.mopEquiv C).functor = CategoryTheory.mopFunctor C
-- CategoryTheory.MonoidalOpposite.mopEquiv_unitIso
@[simps] def MonoidalOpposite.mopEquiv : C ≌ Cᴹᵒᵖ 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C]
Goal: (CategoryTheory.MonoidalOpposite.mopEquiv C).unitIso = CategoryTheory.Iso.refl (𝟭 C)
-- CategoryTheory.MonoidalOpposite.unmopEquiv_unitIso_inv_app_unmop
@[simps!] def MonoidalOpposite.unmopEquiv : Cᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : Cᴹᵒᵖ)
Goal: ((CategoryTheory.MonoidalOpposite.unmopEquiv C).unitIso.inv.app X).unmop = 𝟙 X.unmop
-- CategoryTheory.MonoidalOpposite.unmopEquiv_inverse_obj_unmop
@[simps!] def MonoidalOpposite.unmopEquiv : Cᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (unmop : C)
Goal: ((CategoryTheory.MonoidalOpposite.unmopEquiv C).inverse.obj unmop).unmop = unmop
-- CategoryTheory.MonoidalOpposite.unmopEquiv_inverse_map_unmop
@[simps!] def MonoidalOpposite.unmopEquiv : Cᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C]
Goal: ∀ {X Y : C} (f : X ⟶ Y), ((CategoryTheory.MonoidalOpposite.unmopEquiv C).inverse.map f).unmop = f
-- CategoryTheory.MonoidalOpposite.unmopEquiv_functor_map
@[simps!] def MonoidalOpposite.unmopEquiv : Cᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C]
Goal: ∀ {X Y : Cᴹᵒᵖ} (f : X ⟶ Y), (CategoryTheory.MonoidalOpposite.unmopEquiv C).functor.map f = f.unmop
-- CategoryTheory.MonoidalOpposite.unmopEquiv_counitIso_inv_app
@[simps!] def MonoidalOpposite.unmopEquiv : Cᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : C)
Goal: (CategoryTheory.MonoidalOpposite.unmopEquiv C).counitIso.inv.app X = 𝟙 X
-- CategoryTheory.MonoidalOpposite.unmopEquiv_unitIso_hom_app_unmop
@[simps!] def MonoidalOpposite.unmopEquiv : Cᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : Cᴹᵒᵖ)
Goal: ((CategoryTheory.MonoidalOpposite.unmopEquiv C).unitIso.hom.app X).unmop = 𝟙 X.unmop
-- CategoryTheory.MonoidalOpposite.unmopEquiv_functor_obj
@[simps!] def MonoidalOpposite.unmopEquiv : Cᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (self : Cᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalOpposite.unmopEquiv C).functor.obj self = self.unmop
-- CategoryTheory.MonoidalOpposite.unmopEquiv_counitIso_hom_app
@[simps!] def MonoidalOpposite.unmopEquiv : Cᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : C)
Goal: (CategoryTheory.MonoidalOpposite.unmopEquiv C).counitIso.hom.app X = 𝟙 X
-- CategoryTheory.MonoidalOpposite.mopMopEquivalence_functor_map
@[simps!] def MonoidalOpposite.mopMopEquivalence : Cᴹᵒᵖᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C]
Goal: ∀ {X Y : Cᴹᵒᵖᴹᵒᵖ} (f : X ⟶ Y), (CategoryTheory.MonoidalOpposite.mopMopEquivalence C).functor.map f = f.unmop.unmop
-- CategoryTheory.MonoidalOpposite.mopMopEquivalence_unitIso_hom_app_unmop_unmop
@[simps!] def MonoidalOpposite.mopMopEquivalence : Cᴹᵒᵖᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : Cᴹᵒᵖᴹᵒᵖ)
Goal: ((CategoryTheory.MonoidalOpposite.mopMopEquivalence C).unitIso.hom.app X).unmop.unmop = (𝟙 X.unmop).unmop
-- CategoryTheory.MonoidalOpposite.mopMopEquivalence_counitIso_hom_app
@[simps!] def MonoidalOpposite.mopMopEquivalence : Cᴹᵒᵖᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : C)
Goal: (CategoryTheory.MonoidalOpposite.mopMopEquivalence C).counitIso.hom.app X = 𝟙 X
-- CategoryTheory.MonoidalOpposite.mopMopEquivalence_inverse_obj_unmop_unmop
@[simps!] def MonoidalOpposite.mopMopEquivalence : Cᴹᵒᵖᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : C)
Goal: ((CategoryTheory.MonoidalOpposite.mopMopEquivalence C).inverse.obj X).unmop.unmop = X
-- CategoryTheory.MonoidalOpposite.mopMopEquivalence_inverse_map_unmop_unmop
@[simps!] def MonoidalOpposite.mopMopEquivalence : Cᴹᵒᵖᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C]
Goal: ∀ {X Y : C} (f : X ⟶ Y), ((CategoryTheory.MonoidalOpposite.mopMopEquivalence C).inverse.map f).unmop.unmop = f
-- CategoryTheory.MonoidalOpposite.mopMopEquivalence_counitIso_inv_app
@[simps!] def MonoidalOpposite.mopMopEquivalence : Cᴹᵒᵖᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : C)
Goal: (CategoryTheory.MonoidalOpposite.mopMopEquivalence C).counitIso.inv.app X = 𝟙 X
-- CategoryTheory.MonoidalOpposite.mopMopEquivalence_unitIso_inv_app_unmop_unmop
@[simps!] def MonoidalOpposite.mopMopEquivalence : Cᴹᵒᵖᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : Cᴹᵒᵖᴹᵒᵖ)
Goal: ((CategoryTheory.MonoidalOpposite.mopMopEquivalence C).unitIso.inv.app X).unmop.unmop = (𝟙 X.unmop).unmop
-- CategoryTheory.MonoidalOpposite.mopMopEquivalence_functor_obj
@[simps!] def MonoidalOpposite.mopMopEquivalence : Cᴹᵒᵖᴹᵒᵖ ≌ C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] (X : Cᴹᵒᵖᴹᵒᵖ)
Goal: (CategoryTheory.MonoidalOpposite.mopMopEquivalence C).functor.obj X = X.unmop.unmop
-- CategoryTheory.MonoidalOpposite.tensorIso_hom_app_unmop
@[simps!]
def MonoidalOpposite.tensorIso :
    tensor Cᴹᵒᵖ ≅ (unmopFunctor C).prod (unmopFunctor C) ⋙
      Prod.swap C C ⋙ tensor C ⋙ mopFunctor C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ × Cᴹᵒᵖ)
Goal: ((CategoryTheory.MonoidalOpposite.tensorIso C).hom.app X).unmop =
  𝟙 (CategoryTheory.MonoidalCategory.tensorObj X.2.unmop X.1.unmop)
-- CategoryTheory.MonoidalOpposite.tensorIso_inv_app_unmop
@[simps!]
def MonoidalOpposite.tensorIso :
    tensor Cᴹᵒᵖ ≅ (unmopFunctor C).prod (unmopFunctor C) ⋙
      Prod.swap C C ⋙ tensor C ⋙ mopFunctor C 
Arguments: (C : Type u₁) [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ × Cᴹᵒᵖ)
Goal: ((CategoryTheory.MonoidalOpposite.tensorIso C).inv.app X).unmop =
  𝟙 (CategoryTheory.MonoidalCategory.tensorObj X.2.unmop X.1.unmop)
-- CategoryTheory.MonoidalOpposite.tensorLeftIso_hom_app_unmop
@[simps!]
def MonoidalOpposite.tensorLeftIso (X : Cᴹᵒᵖ) :
    tensorLeft X ≅ unmopFunctor C ⋙ tensorRight (unmop X) ⋙ mopFunctor C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: ∀ (X_1 : Cᴹᵒᵖ), (X.tensorLeftIso.hom.app X_1).unmop = 𝟙 (CategoryTheory.MonoidalCategory.tensorObj X_1.unmop X.unmop)
-- CategoryTheory.MonoidalOpposite.tensorLeftIso_inv_app_unmop
@[simps!]
def MonoidalOpposite.tensorLeftIso (X : Cᴹᵒᵖ) :
    tensorLeft X ≅ unmopFunctor C ⋙ tensorRight (unmop X) ⋙ mopFunctor C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: ∀ (X_1 : Cᴹᵒᵖ), (X.tensorLeftIso.inv.app X_1).unmop = 𝟙 (CategoryTheory.MonoidalCategory.tensorObj X_1.unmop X.unmop)
-- CategoryTheory.MonoidalOpposite.tensorLeftMopIso_hom_app_unmop
@[simps!]
def MonoidalOpposite.tensorLeftMopIso (X : C) :
    tensorLeft (mop X) ≅ unmopFunctor C ⋙ tensorRight X ⋙ mopFunctor C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: ∀ (X_1 : Cᴹᵒᵖ),
  ((CategoryTheory.MonoidalOpposite.tensorLeftMopIso X).hom.app X_1).unmop =
    𝟙 (CategoryTheory.MonoidalCategory.tensorObj X_1.unmop X)
-- CategoryTheory.MonoidalOpposite.tensorLeftMopIso_inv_app_unmop
@[simps!]
def MonoidalOpposite.tensorLeftMopIso (X : C) :
    tensorLeft (mop X) ≅ unmopFunctor C ⋙ tensorRight X ⋙ mopFunctor C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : C)
Goal: ∀ (X_1 : Cᴹᵒᵖ),
  ((CategoryTheory.MonoidalOpposite.tensorLeftMopIso X).inv.app X_1).unmop =
    𝟙 (CategoryTheory.MonoidalCategory.tensorObj X_1.unmop X)
-- CategoryTheory.MonoidalOpposite.tensorLeftUnmopIso_hom_app
@[simps!]
def MonoidalOpposite.tensorLeftUnmopIso (X : Cᴹᵒᵖ) :
    tensorLeft (unmop X) ≅ mopFunctor C ⋙ tensorRight X ⋙ unmopFunctor C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: ∀ (X_1 : C), X.tensorLeftUnmopIso.hom.app X_1 = 𝟙 (CategoryTheory.MonoidalCategory.tensorObj X.unmop X_1)
-- CategoryTheory.MonoidalOpposite.tensorLeftUnmopIso_inv_app
@[simps!]
def MonoidalOpposite.tensorLeftUnmopIso (X : Cᴹᵒᵖ) :
    tensorLeft (unmop X) ≅ mopFunctor C ⋙ tensorRight X ⋙ unmopFunctor C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: ∀ (X_1 : C), X.tensorLeftUnmopIso.inv.app X_1 = 𝟙 (CategoryTheory.MonoidalCategory.tensorObj X.unmop X_1)
-- CategoryTheory.MonoidalOpposite.tensorRightIso_inv_app_unmop
@[simps!]
def MonoidalOpposite.tensorRightIso (X : Cᴹᵒᵖ) :
    tensorRight X ≅ unmopFunctor C ⋙ tensorLeft (unmop X) ⋙ mopFunctor C 
Arguments: {C : Type u₁} [CategoryTheory.Category.{v₁, u₁} C] [CategoryTheory.MonoidalCategory C] (X : Cᴹᵒᵖ)
Goal: ∀ (X_1 : Cᴹᵒᵖ), (X.tensorRightIso.inv.app X_1).unmop = 𝟙 (CategoryTheory.MonoidalCategory.tensorObj X.unmop X_1.unmop)