category_theory.monoidal.braided ⟷ Mathlib.CategoryTheory.Monoidal.Braided.Basic

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -97,8 +97,8 @@ Verifying the axioms for a braiding by checking that the candidate braiding is s
 by a faithful monoidal functor.
 -/
 def braidedCategoryOfFaithful {C D : Type _} [Category C] [Category D] [MonoidalCategory C]
-    [MonoidalCategory D] (F : MonoidalFunctor C D) [Faithful F.toFunctor] [BraidedCategory D]
-    (β : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X)
+    [MonoidalCategory D] (F : MonoidalFunctor C D) [CategoryTheory.Functor.Faithful F.toFunctor]
+    [BraidedCategory D] (β : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X)
     (w : ∀ X Y, F.μ _ _ ≫ F.map (β X Y).Hom = (β_ _ _).Hom ≫ F.μ _ _) : BraidedCategory C
     where
   braiding := β
@@ -136,8 +136,9 @@ def braidedCategoryOfFaithful {C D : Type _} [Category C] [Category D] [Monoidal
 #print CategoryTheory.braidedCategoryOfFullyFaithful /-
 /-- Pull back a braiding along a fully faithful monoidal functor. -/
 noncomputable def braidedCategoryOfFullyFaithful {C D : Type _} [Category C] [Category D]
-    [MonoidalCategory C] [MonoidalCategory D] (F : MonoidalFunctor C D) [Full F.toFunctor]
-    [Faithful F.toFunctor] [BraidedCategory D] : BraidedCategory C :=
+    [MonoidalCategory C] [MonoidalCategory D] (F : MonoidalFunctor C D)
+    [CategoryTheory.Functor.Full F.toFunctor] [CategoryTheory.Functor.Faithful F.toFunctor]
+    [BraidedCategory D] : BraidedCategory C :=
   braidedCategoryOfFaithful F
     (fun X Y =>
       F.toFunctor.preimageIso ((asIso (F.μ _ _)).symm ≪≫ β_ (F.obj X) (F.obj Y) ≪≫ asIso (F.μ _ _)))
@@ -420,7 +421,7 @@ attribute [simp] braided_functor.braided
 /-- A braided category with a braided functor to a symmetric category is itself symmetric. -/
 def symmetricCategoryOfFaithful {C D : Type _} [Category C] [Category D] [MonoidalCategory C]
     [MonoidalCategory D] [BraidedCategory C] [SymmetricCategory D] (F : BraidedFunctor C D)
-    [Faithful F.toFunctor] : SymmetricCategory C
+    [CategoryTheory.Functor.Faithful F.toFunctor] : SymmetricCategory C
     where symmetry' X Y := F.toFunctor.map_injective (by simp)
 #align category_theory.symmetric_category_of_faithful CategoryTheory.symmetricCategoryOfFaithful
 -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/65a1391a0106c9204fe45bc73a039f056558cb83

@@ -528,13 +528,11 @@ def tensor_μ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-#print CategoryTheory.tensor_μ_def₁ /-
 theorem tensor_μ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
     tensor_μ C (X₁, X₂) (Y₁, Y₂) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) =
       (α_ X₁ X₂ (Y₁ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ≫ (𝟙 X₁ ⊗ (β_ X₂ Y₁).Hom ⊗ 𝟙 Y₂) :=
   by dsimp [tensor_μ]; simp
 #align category_theory.tensor_μ_def₁ CategoryTheory.tensor_μ_def₁
--/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -542,13 +540,11 @@ theorem tensor_μ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-#print CategoryTheory.tensor_μ_def₂ /-
 theorem tensor_μ_def₂ (X₁ X₂ Y₁ Y₂ : C) :
     (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).Hom) ≫ (α_ X₁ X₂ (Y₁ ⊗ Y₂)).inv ≫ tensor_μ C (X₁, X₂) (Y₁, Y₂) =
       (𝟙 X₁ ⊗ (β_ X₂ Y₁).Hom ⊗ 𝟙 Y₂) ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).Hom) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv :=
   by dsimp [tensor_μ]; simp
 #align category_theory.tensor_μ_def₂ CategoryTheory.tensor_μ_def₂
--/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -648,7 +644,6 @@ theorem tensor_right_unitality (X₁ X₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-#print CategoryTheory.tensor_associativity_aux /-
 /-
 Diagram B6 from Proposition 1 of [Joyal and Street, *Braided monoidal categories*][Joyal_Street].
 -/
@@ -669,7 +664,6 @@ theorem tensor_associativity_aux (W X Y Z : C) :
   slice_rhs 2 3 => rw [← tensor_id, associator_naturality]
   slice_rhs 3 5 => rw [← tensor_comp, ← tensor_comp, ← hexagon_reverse, tensor_comp, tensor_comp]
 #align category_theory.tensor_associativity_aux CategoryTheory.tensor_associativity_aux
--/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -965,7 +959,6 @@ theorem rightUnitor_monoidal (X₁ X₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-#print CategoryTheory.associator_monoidal_aux /-
 theorem associator_monoidal_aux (W X Y Z : C) :
     (𝟙 W ⊗ (β_ X (Y ⊗ Z)).Hom) ≫
         (𝟙 W ⊗ (α_ Y Z X).Hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).Hom ⊗ 𝟙 (Z ⊗ X)) =
@@ -984,7 +977,6 @@ theorem associator_monoidal_aux (W X Y Z : C) :
   slice_rhs 2 4 => rw [← tensor_comp, ← tensor_comp, ← hexagon_forward, tensor_comp, tensor_comp]
   simp
 #align category_theory.associator_monoidal_aux CategoryTheory.associator_monoidal_aux
--/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/ce64cd319bb6b3e82f31c2d38e79080d377be451

@@ -3,9 +3,9 @@ Copyright (c) 2020 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
 -/
-import Mathbin.CategoryTheory.Monoidal.CoherenceLemmas
-import Mathbin.CategoryTheory.Monoidal.NaturalTransformation
-import Mathbin.CategoryTheory.Monoidal.Discrete
+import CategoryTheory.Monoidal.CoherenceLemmas
+import CategoryTheory.Monoidal.NaturalTransformation
+import CategoryTheory.Monoidal.Discrete
 
 #align_import category_theory.monoidal.braided from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/442a83d738cb208d3600056c489be16900ba701d

@@ -75,14 +75,8 @@ class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] whe
 #align category_theory.braided_category CategoryTheory.BraidedCategory
 -/
 
-restate_axiom braided_category.braiding_naturality'
-
 attribute [simp, reassoc] braided_category.braiding_naturality
 
-restate_axiom braided_category.hexagon_forward'
-
-restate_axiom braided_category.hexagon_reverse'
-
 attribute [reassoc] braided_category.hexagon_forward braided_category.hexagon_reverse
 
 open Category
@@ -331,8 +325,6 @@ class SymmetricCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] e
 #align category_theory.symmetric_category CategoryTheory.SymmetricCategory
 -/
 
-restate_axiom symmetric_category.symmetry'
-
 attribute [simp, reassoc] symmetric_category.symmetry
 
 initialize_simps_projections SymmetricCategory (to_braided_category_braiding → braiding,
@@ -353,8 +345,6 @@ structure LaxBraidedFunctor extends LaxMonoidalFunctor C D where
 #align category_theory.lax_braided_functor CategoryTheory.LaxBraidedFunctor
 -/
 
-restate_axiom lax_braided_functor.braided'
-
 namespace LaxBraidedFunctor
 
 #print CategoryTheory.LaxBraidedFunctor.id /-
@@ -424,8 +414,6 @@ structure BraidedFunctor extends MonoidalFunctor C D where
 #align category_theory.braided_functor CategoryTheory.BraidedFunctor
 -/
 
-restate_axiom braided_functor.braided'
-
 attribute [simp] braided_functor.braided
 
 #print CategoryTheory.symmetricCategoryOfFaithful /-

mathlib commit https://github.com/leanprover-community/mathlib/commit/8ea5598db6caeddde6cb734aa179cc2408dbd345

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module category_theory.monoidal.braided
-! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathbin.CategoryTheory.Monoidal.CoherenceLemmas
 import Mathbin.CategoryTheory.Monoidal.NaturalTransformation
 import Mathbin.CategoryTheory.Monoidal.Discrete
 
+#align_import category_theory.monoidal.braided from "leanprover-community/mathlib"@"af471b9e3ce868f296626d33189b4ce730fa4c00"
+
 /-!
 # Braided and symmetric monoidal categories
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/9fb8964792b4237dac6200193a0d533f1b3f7423

@@ -94,13 +94,13 @@ open MonoidalCategory
 
 open BraidedCategory
 
--- mathport name: exprβ_
 notation "β_" => braiding
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.braidedCategoryOfFaithful /-
 /--
 Verifying the axioms for a braiding by checking that the candidate braiding is sent to a braiding
 by a faithful monoidal functor.
@@ -140,6 +140,7 @@ def braidedCategoryOfFaithful {C D : Type _} [Category C] [Category D] [Monoidal
       Functor.map_id, ← comp_tensor_id_assoc, w, comp_tensor_id_assoc, reassoc_of w,
       braiding_naturality_assoc, lax_monoidal_functor.associativity_inv, hexagon_reverse_assoc]
 #align category_theory.braided_category_of_faithful CategoryTheory.braidedCategoryOfFaithful
+-/
 
 #print CategoryTheory.braidedCategoryOfFullyFaithful /-
 /-- Pull back a braiding along a fully faithful monoidal functor. -/
@@ -504,6 +505,7 @@ instance : BraidedCategory (Discrete M) where braiding X Y := Discrete.eqToIso (
 
 variable {M} {N : Type u} [CommMonoid N]
 
+#print CategoryTheory.Discrete.braidedFunctor /-
 /-- A multiplicative morphism between commutative monoids gives a braided functor between
 the corresponding discrete braided monoidal categories.
 -/
@@ -511,6 +513,7 @@ the corresponding discrete braided monoidal categories.
 def Discrete.braidedFunctor (F : M →* N) : BraidedFunctor (Discrete M) (Discrete N) :=
   { Discrete.monoidalFunctor F with }
 #align category_theory.discrete.braided_functor CategoryTheory.Discrete.braidedFunctor
+-/
 
 end CommMonoid
 
@@ -524,6 +527,7 @@ section Tensor
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.tensor_μ /-
 /-- The strength of the tensor product functor from `C × C` to `C`. -/
 def tensor_μ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor C).obj (X ⊗ Y) :=
   (α_ X.1 X.2 (Y.1 ⊗ Y.2)).Hom ≫
@@ -531,6 +535,7 @@ def tensor_μ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor
       (𝟙 X.1 ⊗ (β_ X.2 Y.1).Hom ⊗ 𝟙 Y.2) ≫
         (𝟙 X.1 ⊗ (α_ Y.1 X.2 Y.2).Hom) ≫ (α_ X.1 Y.1 (X.2 ⊗ Y.2)).inv
 #align category_theory.tensor_μ CategoryTheory.tensor_μ
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -538,11 +543,13 @@ def tensor_μ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.tensor_μ_def₁ /-
 theorem tensor_μ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
     tensor_μ C (X₁, X₂) (Y₁, Y₂) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) =
       (α_ X₁ X₂ (Y₁ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ≫ (𝟙 X₁ ⊗ (β_ X₂ Y₁).Hom ⊗ 𝟙 Y₂) :=
   by dsimp [tensor_μ]; simp
 #align category_theory.tensor_μ_def₁ CategoryTheory.tensor_μ_def₁
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -550,11 +557,13 @@ theorem tensor_μ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.tensor_μ_def₂ /-
 theorem tensor_μ_def₂ (X₁ X₂ Y₁ Y₂ : C) :
     (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).Hom) ≫ (α_ X₁ X₂ (Y₁ ⊗ Y₂)).inv ≫ tensor_μ C (X₁, X₂) (Y₁, Y₂) =
       (𝟙 X₁ ⊗ (β_ X₂ Y₁).Hom ⊗ 𝟙 Y₂) ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).Hom) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv :=
   by dsimp [tensor_μ]; simp
 #align category_theory.tensor_μ_def₂ CategoryTheory.tensor_μ_def₂
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -562,6 +571,7 @@ theorem tensor_μ_def₂ (X₁ X₂ Y₁ Y₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.tensor_μ_natural /-
 theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : U₁ ⟶ V₁)
     (g₂ : U₂ ⟶ V₂) :
     ((f₁ ⊗ f₂) ⊗ g₁ ⊗ g₂) ≫ tensor_μ C (Y₁, Y₂) (V₁, V₂) =
@@ -578,6 +588,7 @@ theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ :
   slice_lhs 5 6 => rw [associator_inv_naturality]
   simp only [assoc]
 #align category_theory.tensor_μ_natural CategoryTheory.tensor_μ_natural
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -589,6 +600,7 @@ theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.tensor_left_unitality /-
 theorem tensor_left_unitality (X₁ X₂ : C) :
     (λ_ (X₁ ⊗ X₂)).Hom =
       ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (X₁ ⊗ X₂)) ≫
@@ -605,6 +617,7 @@ theorem tensor_left_unitality (X₁ X₂ : C) :
   simp only [assoc]
   coherence
 #align category_theory.tensor_left_unitality CategoryTheory.tensor_left_unitality
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -616,6 +629,7 @@ theorem tensor_left_unitality (X₁ X₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.tensor_right_unitality /-
 theorem tensor_right_unitality (X₁ X₂ : C) :
     (ρ_ (X₁ ⊗ X₂)).Hom =
       (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).inv) ≫
@@ -632,6 +646,7 @@ theorem tensor_right_unitality (X₁ X₂ : C) :
   simp only [assoc]
   coherence
 #align category_theory.tensor_right_unitality CategoryTheory.tensor_right_unitality
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -792,6 +807,7 @@ theorem tensor_associativity_aux (W X Y Z : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.tensor_associativity /-
 theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
     (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫
         tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).Hom ⊗ (α_ X₂ Y₂ Z₂).Hom) =
@@ -875,6 +891,7 @@ theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
   dsimp
   coherence
 #align category_theory.tensor_associativity CategoryTheory.tensor_associativity
+-/
 
 #print CategoryTheory.tensorMonoidal /-
 /-- The tensor product functor from `C × C` to `C` as a monoidal functor. -/
@@ -901,6 +918,7 @@ def tensorMonoidal : MonoidalFunctor (C × C) C :=
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.leftUnitor_monoidal /-
 theorem leftUnitor_monoidal (X₁ X₂ : C) :
     (λ_ X₁).Hom ⊗ (λ_ X₂).Hom =
       tensor_μ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ ((λ_ (𝟙_ C)).Hom ⊗ 𝟙 (X₁ ⊗ X₂)) ≫ (λ_ (X₁ ⊗ X₂)).Hom :=
@@ -917,6 +935,7 @@ theorem leftUnitor_monoidal (X₁ X₂ : C) :
   slice_lhs 3 4 => rw [← left_unitor_naturality]
   coherence
 #align category_theory.left_unitor_monoidal CategoryTheory.leftUnitor_monoidal
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -928,6 +947,7 @@ theorem leftUnitor_monoidal (X₁ X₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.rightUnitor_monoidal /-
 theorem rightUnitor_monoidal (X₁ X₂ : C) :
     (ρ_ X₁).Hom ⊗ (ρ_ X₂).Hom =
       tensor_μ C (X₁, 𝟙_ C) (X₂, 𝟙_ C) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).Hom) ≫ (ρ_ (X₁ ⊗ X₂)).Hom :=
@@ -944,6 +964,7 @@ theorem rightUnitor_monoidal (X₁ X₂ : C) :
   slice_lhs 3 4 => rw [← tensor_comp, ← right_unitor_naturality, tensor_comp]
   coherence
 #align category_theory.right_unitor_monoidal CategoryTheory.rightUnitor_monoidal
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1100,6 +1121,7 @@ theorem associator_monoidal_aux (W X Y Z : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.associator_monoidal /-
 theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ : C) :
     tensor_μ C (X₁ ⊗ X₂, X₃) (Y₁ ⊗ Y₂, Y₃) ≫
         (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).Hom =
@@ -1176,6 +1198,7 @@ theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ : C) :
   dsimp
   coherence
 #align category_theory.associator_monoidal CategoryTheory.associator_monoidal
+-/
 
 end Tensor
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/7e5137f579de09a059a5ce98f364a04e221aabf0

@@ -225,7 +225,6 @@ theorem braiding_leftUnitor_aux₂ (X : C) :
       slice_lhs 2 3 => rw [← braiding_naturality]; simp only [assoc]
     _ = (α_ _ _ _).Hom ≫ (𝟙 _ ⊗ (λ_ _).Hom) := by rw [iso.hom_inv_id, comp_id]
     _ = (ρ_ X).Hom ⊗ 𝟙 (𝟙_ C) := by rw [triangle]
-    
 #align category_theory.braiding_left_unitor_aux₂ CategoryTheory.braiding_leftUnitor_aux₂
 -/
 
@@ -290,7 +289,6 @@ theorem braiding_rightUnitor_aux₂ (X : C) :
       slice_lhs 2 3 => rw [← braiding_naturality]; simp only [assoc]
     _ = (α_ _ _ _).inv ≫ ((ρ_ _).Hom ⊗ 𝟙 _) := by rw [iso.hom_inv_id, comp_id]
     _ = 𝟙 (𝟙_ C) ⊗ (λ_ X).Hom := by rw [triangle_assoc_comp_right]
-    
 #align category_theory.braiding_right_unitor_aux₂ CategoryTheory.braiding_rightUnitor_aux₂
 -/
 

mathlib commit https://github.com/leanprover-community/mathlib/commit/5f25c089cb34db4db112556f23c50d12da81b297

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
 
 ! This file was ported from Lean 3 source module category_theory.monoidal.braided
-! leanprover-community/mathlib commit 2efd2423f8d25fa57cf7a179f5d8652ab4d0df44
+! leanprover-community/mathlib commit af471b9e3ce868f296626d33189b4ce730fa4c00
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -15,6 +15,9 @@ import Mathbin.CategoryTheory.Monoidal.Discrete
 /-!
 # Braided and symmetric monoidal categories
 
+> THIS FILE IS SYNCHRONIZED WITH MATHLIB4.
+> Any changes to this file require a corresponding PR to mathlib4.
+
 The basic definitions of braided monoidal categories, and symmetric monoidal categories,
 as well as braided functors.
 
@@ -38,6 +41,7 @@ universe v v₁ v₂ v₃ u u₁ u₂ u₃
 
 namespace CategoryTheory
 
+#print CategoryTheory.BraidedCategory /-
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -72,6 +76,7 @@ class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] whe
         (𝟙 X ⊗ (braiding Y Z).Hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).Hom ⊗ 𝟙 Y) := by
     obviously
 #align category_theory.braided_category CategoryTheory.BraidedCategory
+-/
 
 restate_axiom braided_category.braiding_naturality'
 
@@ -136,6 +141,7 @@ def braidedCategoryOfFaithful {C D : Type _} [Category C] [Category D] [Monoidal
       braiding_naturality_assoc, lax_monoidal_functor.associativity_inv, hexagon_reverse_assoc]
 #align category_theory.braided_category_of_faithful CategoryTheory.braidedCategoryOfFaithful
 
+#print CategoryTheory.braidedCategoryOfFullyFaithful /-
 /-- Pull back a braiding along a fully faithful monoidal functor. -/
 noncomputable def braidedCategoryOfFullyFaithful {C D : Type _} [Category C] [Category D]
     [MonoidalCategory C] [MonoidalCategory D] (F : MonoidalFunctor C D) [Full F.toFunctor]
@@ -145,6 +151,7 @@ noncomputable def braidedCategoryOfFullyFaithful {C D : Type _} [Category C] [Ca
       F.toFunctor.preimageIso ((asIso (F.μ _ _)).symm ≪≫ β_ (F.obj X) (F.obj Y) ≪≫ asIso (F.μ _ _)))
     (by tidy)
 #align category_theory.braided_category_of_fully_faithful CategoryTheory.braidedCategoryOfFullyFaithful
+-/
 
 section
 
@@ -167,12 +174,14 @@ variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory C] [BraidedCatego
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.braiding_leftUnitor_aux₁ /-
 theorem braiding_leftUnitor_aux₁ (X : C) :
     (α_ (𝟙_ C) (𝟙_ C) X).Hom ≫
         (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).Hom ⊗ 𝟙 _) =
       ((λ_ _).Hom ⊗ 𝟙 X) ≫ (β_ X (𝟙_ C)).inv :=
   by rw [← left_unitor_tensor, left_unitor_naturality]; simp
 #align category_theory.braiding_left_unitor_aux₁ CategoryTheory.braiding_leftUnitor_aux₁
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -191,6 +200,7 @@ theorem braiding_leftUnitor_aux₁ (X : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.braiding_leftUnitor_aux₂ /-
 theorem braiding_leftUnitor_aux₂ (X : C) :
     ((β_ X (𝟙_ C)).Hom ⊗ 𝟙 (𝟙_ C)) ≫ ((λ_ X).Hom ⊗ 𝟙 (𝟙_ C)) = (ρ_ X).Hom ⊗ 𝟙 (𝟙_ C) :=
   calc
@@ -217,21 +227,26 @@ theorem braiding_leftUnitor_aux₂ (X : C) :
     _ = (ρ_ X).Hom ⊗ 𝟙 (𝟙_ C) := by rw [triangle]
     
 #align category_theory.braiding_left_unitor_aux₂ CategoryTheory.braiding_leftUnitor_aux₂
+-/
 
+#print CategoryTheory.braiding_leftUnitor /-
 @[simp]
 theorem braiding_leftUnitor (X : C) : (β_ X (𝟙_ C)).Hom ≫ (λ_ X).Hom = (ρ_ X).Hom := by
   rw [← tensor_right_iff, comp_tensor_id, braiding_left_unitor_aux₂]
 #align category_theory.braiding_left_unitor CategoryTheory.braiding_leftUnitor
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.braiding_rightUnitor_aux₁ /-
 theorem braiding_rightUnitor_aux₁ (X : C) :
     (α_ X (𝟙_ C) (𝟙_ C)).inv ≫
         ((β_ (𝟙_ C) X).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ _ X _).Hom ≫ (𝟙 _ ⊗ (ρ_ X).Hom) =
       (𝟙 X ⊗ (ρ_ _).Hom) ≫ (β_ (𝟙_ C) X).inv :=
   by rw [← right_unitor_tensor, right_unitor_naturality]; simp
 #align category_theory.braiding_right_unitor_aux₁ CategoryTheory.braiding_rightUnitor_aux₁
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -250,6 +265,7 @@ theorem braiding_rightUnitor_aux₁ (X : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.braiding_rightUnitor_aux₂ /-
 theorem braiding_rightUnitor_aux₂ (X : C) :
     (𝟙 (𝟙_ C) ⊗ (β_ (𝟙_ C) X).Hom) ≫ (𝟙 (𝟙_ C) ⊗ (ρ_ X).Hom) = 𝟙 (𝟙_ C) ⊗ (λ_ X).Hom :=
   calc
@@ -276,28 +292,36 @@ theorem braiding_rightUnitor_aux₂ (X : C) :
     _ = 𝟙 (𝟙_ C) ⊗ (λ_ X).Hom := by rw [triangle_assoc_comp_right]
     
 #align category_theory.braiding_right_unitor_aux₂ CategoryTheory.braiding_rightUnitor_aux₂
+-/
 
+#print CategoryTheory.braiding_rightUnitor /-
 @[simp]
 theorem braiding_rightUnitor (X : C) : (β_ (𝟙_ C) X).Hom ≫ (ρ_ X).Hom = (λ_ X).Hom := by
   rw [← tensor_left_iff, id_tensor_comp, braiding_right_unitor_aux₂]
 #align category_theory.braiding_right_unitor CategoryTheory.braiding_rightUnitor
+-/
 
+#print CategoryTheory.leftUnitor_inv_braiding /-
 @[simp]
 theorem leftUnitor_inv_braiding (X : C) : (λ_ X).inv ≫ (β_ (𝟙_ C) X).Hom = (ρ_ X).inv :=
   by
   apply (cancel_mono (ρ_ X).Hom).1
   simp only [assoc, braiding_right_unitor, iso.inv_hom_id]
 #align category_theory.left_unitor_inv_braiding CategoryTheory.leftUnitor_inv_braiding
+-/
 
+#print CategoryTheory.rightUnitor_inv_braiding /-
 @[simp]
 theorem rightUnitor_inv_braiding (X : C) : (ρ_ X).inv ≫ (β_ X (𝟙_ C)).Hom = (λ_ X).inv :=
   by
   apply (cancel_mono (λ_ X).Hom).1
   simp only [assoc, braiding_left_unitor, iso.inv_hom_id]
 #align category_theory.right_unitor_inv_braiding CategoryTheory.rightUnitor_inv_braiding
+-/
 
 end
 
+#print CategoryTheory.SymmetricCategory /-
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /--
 A symmetric monoidal category is a braided monoidal category for which the braiding is symmetric.
@@ -309,6 +333,7 @@ class SymmetricCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] e
   -- braiding symmetric:
   symmetry' : ∀ X Y : C, (β_ X Y).Hom ≫ (β_ Y X).Hom = 𝟙 (X ⊗ Y) := by obviously
 #align category_theory.symmetric_category CategoryTheory.SymmetricCategory
+-/
 
 restate_axiom symmetric_category.symmetry'
 
@@ -323,28 +348,33 @@ variable (D : Type u₂) [Category.{v₂} D] [MonoidalCategory D] [BraidedCatego
 
 variable (E : Type u₃) [Category.{v₃} E] [MonoidalCategory E] [BraidedCategory E]
 
+#print CategoryTheory.LaxBraidedFunctor /-
 /-- A lax braided functor between braided monoidal categories is a lax monoidal functor
 which preserves the braiding.
 -/
 structure LaxBraidedFunctor extends LaxMonoidalFunctor C D where
   braided' : ∀ X Y : C, μ X Y ≫ map (β_ X Y).Hom = (β_ (obj X) (obj Y)).Hom ≫ μ Y X := by obviously
 #align category_theory.lax_braided_functor CategoryTheory.LaxBraidedFunctor
+-/
 
 restate_axiom lax_braided_functor.braided'
 
 namespace LaxBraidedFunctor
 
+#print CategoryTheory.LaxBraidedFunctor.id /-
 /-- The identity lax braided monoidal functor. -/
 @[simps]
 def id : LaxBraidedFunctor C C :=
   { MonoidalFunctor.id C with }
 #align category_theory.lax_braided_functor.id CategoryTheory.LaxBraidedFunctor.id
+-/
 
 instance : Inhabited (LaxBraidedFunctor C C) :=
   ⟨id C⟩
 
 variable {C D E}
 
+#print CategoryTheory.LaxBraidedFunctor.comp /-
 /-- The composition of lax braided monoidal functors. -/
 @[simps]
 def comp (F : LaxBraidedFunctor C D) (G : LaxBraidedFunctor D E) : LaxBraidedFunctor C E :=
@@ -356,17 +386,23 @@ def comp (F : LaxBraidedFunctor C D) (G : LaxBraidedFunctor D E) : LaxBraidedFun
       slice_lhs 1 2 => rw [G.braided]
       simp only [category.assoc] }
 #align category_theory.lax_braided_functor.comp CategoryTheory.LaxBraidedFunctor.comp
+-/
 
+#print CategoryTheory.LaxBraidedFunctor.categoryLaxBraidedFunctor /-
 instance categoryLaxBraidedFunctor : Category (LaxBraidedFunctor C D) :=
   InducedCategory.category LaxBraidedFunctor.toLaxMonoidalFunctor
 #align category_theory.lax_braided_functor.category_lax_braided_functor CategoryTheory.LaxBraidedFunctor.categoryLaxBraidedFunctor
+-/
 
+#print CategoryTheory.LaxBraidedFunctor.comp_toNatTrans /-
 @[simp]
 theorem comp_toNatTrans {F G H : LaxBraidedFunctor C D} {α : F ⟶ G} {β : G ⟶ H} :
     (α ≫ β).toNatTrans = @CategoryStruct.comp (C ⥤ D) _ _ _ _ α.toNatTrans β.toNatTrans :=
   rfl
 #align category_theory.lax_braided_functor.comp_to_nat_trans CategoryTheory.LaxBraidedFunctor.comp_toNatTrans
+-/
 
+#print CategoryTheory.LaxBraidedFunctor.mkIso /-
 /-- Interpret a natural isomorphism of the underlyling lax monoidal functors as an
 isomorphism of the lax braided monoidal functors.
 -/
@@ -375,9 +411,11 @@ def mkIso {F G : LaxBraidedFunctor C D} (i : F.toLaxMonoidalFunctor ≅ G.toLaxM
     F ≅ G :=
   { i with }
 #align category_theory.lax_braided_functor.mk_iso CategoryTheory.LaxBraidedFunctor.mkIso
+-/
 
 end LaxBraidedFunctor
 
+#print CategoryTheory.BraidedFunctor /-
 /-- A braided functor between braided monoidal categories is a monoidal functor
 which preserves the braiding.
 -/
@@ -388,53 +426,67 @@ structure BraidedFunctor extends MonoidalFunctor C D where
   braided' : ∀ X Y : C, map (β_ X Y).Hom = inv (μ X Y) ≫ (β_ (obj X) (obj Y)).Hom ≫ μ Y X := by
     obviously
 #align category_theory.braided_functor CategoryTheory.BraidedFunctor
+-/
 
 restate_axiom braided_functor.braided'
 
 attribute [simp] braided_functor.braided
 
+#print CategoryTheory.symmetricCategoryOfFaithful /-
 /-- A braided category with a braided functor to a symmetric category is itself symmetric. -/
 def symmetricCategoryOfFaithful {C D : Type _} [Category C] [Category D] [MonoidalCategory C]
     [MonoidalCategory D] [BraidedCategory C] [SymmetricCategory D] (F : BraidedFunctor C D)
     [Faithful F.toFunctor] : SymmetricCategory C
     where symmetry' X Y := F.toFunctor.map_injective (by simp)
 #align category_theory.symmetric_category_of_faithful CategoryTheory.symmetricCategoryOfFaithful
+-/
 
 namespace BraidedFunctor
 
+#print CategoryTheory.BraidedFunctor.toLaxBraidedFunctor /-
 /-- Turn a braided functor into a lax braided functor. -/
 @[simps]
 def toLaxBraidedFunctor (F : BraidedFunctor C D) : LaxBraidedFunctor C D :=
   { F with braided' := fun X Y => by rw [F.braided]; simp }
 #align category_theory.braided_functor.to_lax_braided_functor CategoryTheory.BraidedFunctor.toLaxBraidedFunctor
+-/
 
+#print CategoryTheory.BraidedFunctor.id /-
 /-- The identity braided monoidal functor. -/
 @[simps]
 def id : BraidedFunctor C C :=
   { MonoidalFunctor.id C with }
 #align category_theory.braided_functor.id CategoryTheory.BraidedFunctor.id
+-/
 
 instance : Inhabited (BraidedFunctor C C) :=
   ⟨id C⟩
 
 variable {C D E}
 
+#print CategoryTheory.BraidedFunctor.comp /-
 /-- The composition of braided monoidal functors. -/
 @[simps]
 def comp (F : BraidedFunctor C D) (G : BraidedFunctor D E) : BraidedFunctor C E :=
   { MonoidalFunctor.comp F.toMonoidalFunctor G.toMonoidalFunctor with }
 #align category_theory.braided_functor.comp CategoryTheory.BraidedFunctor.comp
+-/
 
+#print CategoryTheory.BraidedFunctor.categoryBraidedFunctor /-
 instance categoryBraidedFunctor : Category (BraidedFunctor C D) :=
   InducedCategory.category BraidedFunctor.toMonoidalFunctor
 #align category_theory.braided_functor.category_braided_functor CategoryTheory.BraidedFunctor.categoryBraidedFunctor
+-/
 
+#print CategoryTheory.BraidedFunctor.comp_toNatTrans /-
 @[simp]
 theorem comp_toNatTrans {F G H : BraidedFunctor C D} {α : F ⟶ G} {β : G ⟶ H} :
     (α ≫ β).toNatTrans = @CategoryStruct.comp (C ⥤ D) _ _ _ _ α.toNatTrans β.toNatTrans :=
   rfl
 #align category_theory.braided_functor.comp_to_nat_trans CategoryTheory.BraidedFunctor.comp_toNatTrans
+-/
 
+#print CategoryTheory.BraidedFunctor.mkIso /-
 /-- Interpret a natural isomorphism of the underlyling monoidal functors as an
 isomorphism of the braided monoidal functors.
 -/
@@ -442,6 +494,7 @@ isomorphism of the braided monoidal functors.
 def mkIso {F G : BraidedFunctor C D} (i : F.toMonoidalFunctor ≅ G.toMonoidalFunctor) : F ≅ G :=
   { i with }
 #align category_theory.braided_functor.mk_iso CategoryTheory.BraidedFunctor.mkIso
+-/
 
 end BraidedFunctor
 
@@ -474,12 +527,12 @@ section Tensor
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /-- The strength of the tensor product functor from `C × C` to `C`. -/
-def tensorμ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor C).obj (X ⊗ Y) :=
+def tensor_μ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor C).obj (X ⊗ Y) :=
   (α_ X.1 X.2 (Y.1 ⊗ Y.2)).Hom ≫
     (𝟙 X.1 ⊗ (α_ X.2 Y.1 Y.2).inv) ≫
       (𝟙 X.1 ⊗ (β_ X.2 Y.1).Hom ⊗ 𝟙 Y.2) ≫
         (𝟙 X.1 ⊗ (α_ Y.1 X.2 Y.2).Hom) ≫ (α_ X.1 Y.1 (X.2 ⊗ Y.2)).inv
-#align category_theory.tensor_μ CategoryTheory.tensorμ
+#align category_theory.tensor_μ CategoryTheory.tensor_μ
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -487,11 +540,11 @@ def tensorμ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-theorem tensorμ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
-    tensorμ C (X₁, X₂) (Y₁, Y₂) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) =
+theorem tensor_μ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
+    tensor_μ C (X₁, X₂) (Y₁, Y₂) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) =
       (α_ X₁ X₂ (Y₁ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ≫ (𝟙 X₁ ⊗ (β_ X₂ Y₁).Hom ⊗ 𝟙 Y₂) :=
   by dsimp [tensor_μ]; simp
-#align category_theory.tensor_μ_def₁ CategoryTheory.tensorμ_def₁
+#align category_theory.tensor_μ_def₁ CategoryTheory.tensor_μ_def₁
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -499,11 +552,11 @@ theorem tensorμ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-theorem tensorμ_def₂ (X₁ X₂ Y₁ Y₂ : C) :
-    (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).Hom) ≫ (α_ X₁ X₂ (Y₁ ⊗ Y₂)).inv ≫ tensorμ C (X₁, X₂) (Y₁, Y₂) =
+theorem tensor_μ_def₂ (X₁ X₂ Y₁ Y₂ : C) :
+    (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).Hom) ≫ (α_ X₁ X₂ (Y₁ ⊗ Y₂)).inv ≫ tensor_μ C (X₁, X₂) (Y₁, Y₂) =
       (𝟙 X₁ ⊗ (β_ X₂ Y₁).Hom ⊗ 𝟙 Y₂) ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).Hom) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv :=
   by dsimp [tensor_μ]; simp
-#align category_theory.tensor_μ_def₂ CategoryTheory.tensorμ_def₂
+#align category_theory.tensor_μ_def₂ CategoryTheory.tensor_μ_def₂
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -511,10 +564,10 @@ theorem tensorμ_def₂ (X₁ X₂ Y₁ Y₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
-theorem tensorμ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : U₁ ⟶ V₁)
+theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : U₁ ⟶ V₁)
     (g₂ : U₂ ⟶ V₂) :
-    ((f₁ ⊗ f₂) ⊗ g₁ ⊗ g₂) ≫ tensorμ C (Y₁, Y₂) (V₁, V₂) =
-      tensorμ C (X₁, X₂) (U₁, U₂) ≫ ((f₁ ⊗ g₁) ⊗ f₂ ⊗ g₂) :=
+    ((f₁ ⊗ f₂) ⊗ g₁ ⊗ g₂) ≫ tensor_μ C (Y₁, Y₂) (V₁, V₂) =
+      tensor_μ C (X₁, X₂) (U₁, U₂) ≫ ((f₁ ⊗ g₁) ⊗ f₂ ⊗ g₂) :=
   by
   dsimp [tensor_μ]
   slice_lhs 1 2 => rw [associator_naturality]
@@ -526,7 +579,7 @@ theorem tensorμ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ : X
   slice_lhs 4 5 => rw [← tensor_comp, comp_id f₁, ← id_comp f₁, associator_naturality, tensor_comp]
   slice_lhs 5 6 => rw [associator_inv_naturality]
   simp only [assoc]
-#align category_theory.tensor_μ_natural CategoryTheory.tensorμ_natural
+#align category_theory.tensor_μ_natural CategoryTheory.tensor_μ_natural
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -541,7 +594,7 @@ theorem tensorμ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ : X
 theorem tensor_left_unitality (X₁ X₂ : C) :
     (λ_ (X₁ ⊗ X₂)).Hom =
       ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (X₁ ⊗ X₂)) ≫
-        tensorμ C (𝟙_ C, 𝟙_ C) (X₁, X₂) ≫ ((λ_ X₁).Hom ⊗ (λ_ X₂).Hom) :=
+        tensor_μ C (𝟙_ C, 𝟙_ C) (X₁, X₂) ≫ ((λ_ X₁).Hom ⊗ (λ_ X₂).Hom) :=
   by
   dsimp [tensor_μ]
   have :
@@ -568,7 +621,7 @@ theorem tensor_left_unitality (X₁ X₂ : C) :
 theorem tensor_right_unitality (X₁ X₂ : C) :
     (ρ_ (X₁ ⊗ X₂)).Hom =
       (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).inv) ≫
-        tensorμ C (X₁, X₂) (𝟙_ C, 𝟙_ C) ≫ ((ρ_ X₁).Hom ⊗ (ρ_ X₂).Hom) :=
+        tensor_μ C (X₁, X₂) (𝟙_ C, 𝟙_ C) ≫ ((ρ_ X₁).Hom ⊗ (ρ_ X₂).Hom) :=
   by
   dsimp [tensor_μ]
   have :
@@ -597,6 +650,7 @@ theorem tensor_right_unitality (X₁ X₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.tensor_associativity_aux /-
 /-
 Diagram B6 from Proposition 1 of [Joyal and Street, *Braided monoidal categories*][Joyal_Street].
 -/
@@ -617,6 +671,7 @@ theorem tensor_associativity_aux (W X Y Z : C) :
   slice_rhs 2 3 => rw [← tensor_id, associator_naturality]
   slice_rhs 3 5 => rw [← tensor_comp, ← tensor_comp, ← hexagon_reverse, tensor_comp, tensor_comp]
 #align category_theory.tensor_associativity_aux CategoryTheory.tensor_associativity_aux
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -740,10 +795,10 @@ theorem tensor_associativity_aux (W X Y Z : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
-    (tensorμ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫
-        tensorμ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).Hom ⊗ (α_ X₂ Y₂ Z₂).Hom) =
+    (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫
+        tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).Hom ⊗ (α_ X₂ Y₂ Z₂).Hom) =
       (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).Hom ≫
-        (𝟙 (X₁ ⊗ X₂) ⊗ tensorμ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensorμ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) :=
+        (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) :=
   by
   have :
     (α_ X₁ Y₁ Z₁).Hom ⊗ (α_ X₂ Y₂ Z₂).Hom =
@@ -823,18 +878,20 @@ theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
   coherence
 #align category_theory.tensor_associativity CategoryTheory.tensor_associativity
 
+#print CategoryTheory.tensorMonoidal /-
 /-- The tensor product functor from `C × C` to `C` as a monoidal functor. -/
 @[simps]
 def tensorMonoidal : MonoidalFunctor (C × C) C :=
   { tensor C with
     ε := (λ_ (𝟙_ C)).inv
-    μ := fun X Y => tensorμ C X Y
-    μ_natural' := fun X Y X' Y' f g => tensorμ_natural C f.1 f.2 g.1 g.2
+    μ := fun X Y => tensor_μ C X Y
+    μ_natural' := fun X Y X' Y' f g => tensor_μ_natural C f.1 f.2 g.1 g.2
     associativity' := fun X Y Z => tensor_associativity C X.1 X.2 Y.1 Y.2 Z.1 Z.2
     left_unitality' := fun ⟨X₁, X₂⟩ => tensor_left_unitality C X₁ X₂
     right_unitality' := fun ⟨X₁, X₂⟩ => tensor_right_unitality C X₁ X₂
     μ_isIso := by dsimp [tensor_μ]; infer_instance }
 #align category_theory.tensor_monoidal CategoryTheory.tensorMonoidal
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -848,7 +905,7 @@ def tensorMonoidal : MonoidalFunctor (C × C) C :=
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem leftUnitor_monoidal (X₁ X₂ : C) :
     (λ_ X₁).Hom ⊗ (λ_ X₂).Hom =
-      tensorμ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ ((λ_ (𝟙_ C)).Hom ⊗ 𝟙 (X₁ ⊗ X₂)) ≫ (λ_ (X₁ ⊗ X₂)).Hom :=
+      tensor_μ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ ((λ_ (𝟙_ C)).Hom ⊗ 𝟙 (X₁ ⊗ X₂)) ≫ (λ_ (X₁ ⊗ X₂)).Hom :=
   by
   dsimp [tensor_μ]
   have :
@@ -875,7 +932,7 @@ theorem leftUnitor_monoidal (X₁ X₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem rightUnitor_monoidal (X₁ X₂ : C) :
     (ρ_ X₁).Hom ⊗ (ρ_ X₂).Hom =
-      tensorμ C (X₁, 𝟙_ C) (X₂, 𝟙_ C) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).Hom) ≫ (ρ_ (X₁ ⊗ X₂)).Hom :=
+      tensor_μ C (X₁, 𝟙_ C) (X₂, 𝟙_ C) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).Hom) ≫ (ρ_ (X₁ ⊗ X₂)).Hom :=
   by
   dsimp [tensor_μ]
   have :
@@ -904,6 +961,7 @@ theorem rightUnitor_monoidal (X₁ X₂ : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
+#print CategoryTheory.associator_monoidal_aux /-
 theorem associator_monoidal_aux (W X Y Z : C) :
     (𝟙 W ⊗ (β_ X (Y ⊗ Z)).Hom) ≫
         (𝟙 W ⊗ (α_ Y Z X).Hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).Hom ⊗ 𝟙 (Z ⊗ X)) =
@@ -922,6 +980,7 @@ theorem associator_monoidal_aux (W X Y Z : C) :
   slice_rhs 2 4 => rw [← tensor_comp, ← tensor_comp, ← hexagon_forward, tensor_comp, tensor_comp]
   simp
 #align category_theory.associator_monoidal_aux CategoryTheory.associator_monoidal_aux
+-/
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -1044,10 +1103,10 @@ theorem associator_monoidal_aux (W X Y Z : C) :
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
 theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ : C) :
-    tensorμ C (X₁ ⊗ X₂, X₃) (Y₁ ⊗ Y₂, Y₃) ≫
-        (tensorμ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).Hom =
+    tensor_μ C (X₁ ⊗ X₂, X₃) (Y₁ ⊗ Y₂, Y₃) ≫
+        (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).Hom =
       ((α_ X₁ X₂ X₃).Hom ⊗ (α_ Y₁ Y₂ Y₃).Hom) ≫
-        tensorμ C (X₁, X₂ ⊗ X₃) (Y₁, Y₂ ⊗ Y₃) ≫ (𝟙 (X₁ ⊗ Y₁) ⊗ tensorμ C (X₂, X₃) (Y₂, Y₃)) :=
+        tensor_μ C (X₁, X₂ ⊗ X₃) (Y₁, Y₂ ⊗ Y₃) ≫ (𝟙 (X₁ ⊗ Y₁) ⊗ tensor_μ C (X₂, X₃) (Y₂, Y₃)) :=
   by
   have :
     (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).Hom =

mathlib commit https://github.com/leanprover-community/mathlib/commit/cca40788df1b8755d5baf17ab2f27dacc2e17acb

@@ -305,7 +305,7 @@ A symmetric monoidal category is a braided monoidal category for which the braid
 See <https://stacks.math.columbia.edu/tag/0FFW>.
 -/
 class SymmetricCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] extends
-  BraidedCategory.{v} C where
+    BraidedCategory.{v} C where
   -- braiding symmetric:
   symmetry' : ∀ X Y : C, (β_ X Y).Hom ≫ (β_ Y X).Hom = 𝟙 (X ⊗ Y) := by obviously
 #align category_theory.symmetric_category CategoryTheory.SymmetricCategory

mathlib commit https://github.com/leanprover-community/mathlib/commit/917c3c072e487b3cccdbfeff17e75b40e45f66cb

@@ -171,9 +171,7 @@ theorem braiding_leftUnitor_aux₁ (X : C) :
     (α_ (𝟙_ C) (𝟙_ C) X).Hom ≫
         (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).Hom ⊗ 𝟙 _) =
       ((λ_ _).Hom ⊗ 𝟙 X) ≫ (β_ X (𝟙_ C)).inv :=
-  by
-  rw [← left_unitor_tensor, left_unitor_naturality]
-  simp
+  by rw [← left_unitor_tensor, left_unitor_naturality]; simp
 #align category_theory.braiding_left_unitor_aux₁ CategoryTheory.braiding_leftUnitor_aux₁
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -205,22 +203,16 @@ theorem braiding_leftUnitor_aux₂ (X : C) :
           (α_ _ _ _).Hom ≫
             (𝟙 _ ⊗ (β_ X _).Hom) ≫
               (𝟙 _ ⊗ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫ ((λ_ X).Hom ⊗ 𝟙 (𝟙_ C)) :=
-      by
-      slice_rhs 3 4 => rw [← id_tensor_comp, iso.hom_inv_id, tensor_id]
-      rw [id_comp]
+      by slice_rhs 3 4 => rw [← id_tensor_comp, iso.hom_inv_id, tensor_id]; rw [id_comp]
     _ =
         (α_ _ _ _).Hom ≫
           (β_ _ _).Hom ≫
             (α_ _ _ _).Hom ≫ (𝟙 _ ⊗ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫ ((λ_ X).Hom ⊗ 𝟙 (𝟙_ C)) :=
-      by
-      slice_lhs 1 3 => rw [← hexagon_forward]
-      simp only [assoc]
+      by slice_lhs 1 3 => rw [← hexagon_forward]; simp only [assoc]
     _ = (α_ _ _ _).Hom ≫ (β_ _ _).Hom ≫ ((λ_ _).Hom ⊗ 𝟙 X) ≫ (β_ X _).inv := by
       rw [braiding_left_unitor_aux₁]
-    _ = (α_ _ _ _).Hom ≫ (𝟙 _ ⊗ (λ_ _).Hom) ≫ (β_ _ _).Hom ≫ (β_ X _).inv :=
-      by
-      slice_lhs 2 3 => rw [← braiding_naturality]
-      simp only [assoc]
+    _ = (α_ _ _ _).Hom ≫ (𝟙 _ ⊗ (λ_ _).Hom) ≫ (β_ _ _).Hom ≫ (β_ X _).inv := by
+      slice_lhs 2 3 => rw [← braiding_naturality]; simp only [assoc]
     _ = (α_ _ _ _).Hom ≫ (𝟙 _ ⊗ (λ_ _).Hom) := by rw [iso.hom_inv_id, comp_id]
     _ = (ρ_ X).Hom ⊗ 𝟙 (𝟙_ C) := by rw [triangle]
     
@@ -238,9 +230,7 @@ theorem braiding_rightUnitor_aux₁ (X : C) :
     (α_ X (𝟙_ C) (𝟙_ C)).inv ≫
         ((β_ (𝟙_ C) X).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ _ X _).Hom ≫ (𝟙 _ ⊗ (ρ_ X).Hom) =
       (𝟙 X ⊗ (ρ_ _).Hom) ≫ (β_ (𝟙_ C) X).inv :=
-  by
-  rw [← right_unitor_tensor, right_unitor_naturality]
-  simp
+  by rw [← right_unitor_tensor, right_unitor_naturality]; simp
 #align category_theory.braiding_right_unitor_aux₁ CategoryTheory.braiding_rightUnitor_aux₁
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -272,22 +262,16 @@ theorem braiding_rightUnitor_aux₂ (X : C) :
           (α_ _ _ _).inv ≫
             ((β_ _ X).Hom ⊗ 𝟙 _) ≫
               ((β_ _ X).inv ⊗ 𝟙 _) ≫ (α_ _ _ _).Hom ≫ (𝟙 (𝟙_ C) ⊗ (ρ_ X).Hom) :=
-      by
-      slice_rhs 3 4 => rw [← comp_tensor_id, iso.hom_inv_id, tensor_id]
-      rw [id_comp]
+      by slice_rhs 3 4 => rw [← comp_tensor_id, iso.hom_inv_id, tensor_id]; rw [id_comp]
     _ =
         (α_ _ _ _).inv ≫
           (β_ _ _).Hom ≫
             (α_ _ _ _).inv ≫ ((β_ _ X).inv ⊗ 𝟙 _) ≫ (α_ _ _ _).Hom ≫ (𝟙 (𝟙_ C) ⊗ (ρ_ X).Hom) :=
-      by
-      slice_lhs 1 3 => rw [← hexagon_reverse]
-      simp only [assoc]
+      by slice_lhs 1 3 => rw [← hexagon_reverse]; simp only [assoc]
     _ = (α_ _ _ _).inv ≫ (β_ _ _).Hom ≫ (𝟙 X ⊗ (ρ_ _).Hom) ≫ (β_ _ X).inv := by
       rw [braiding_right_unitor_aux₁]
-    _ = (α_ _ _ _).inv ≫ ((ρ_ _).Hom ⊗ 𝟙 _) ≫ (β_ _ X).Hom ≫ (β_ _ _).inv :=
-      by
-      slice_lhs 2 3 => rw [← braiding_naturality]
-      simp only [assoc]
+    _ = (α_ _ _ _).inv ≫ ((ρ_ _).Hom ⊗ 𝟙 _) ≫ (β_ _ X).Hom ≫ (β_ _ _).inv := by
+      slice_lhs 2 3 => rw [← braiding_naturality]; simp only [assoc]
     _ = (α_ _ _ _).inv ≫ ((ρ_ _).Hom ⊗ 𝟙 _) := by rw [iso.hom_inv_id, comp_id]
     _ = 𝟙 (𝟙_ C) ⊗ (λ_ X).Hom := by rw [triangle_assoc_comp_right]
     
@@ -421,10 +405,7 @@ namespace BraidedFunctor
 /-- Turn a braided functor into a lax braided functor. -/
 @[simps]
 def toLaxBraidedFunctor (F : BraidedFunctor C D) : LaxBraidedFunctor C D :=
-  { F with
-    braided' := fun X Y => by
-      rw [F.braided]
-      simp }
+  { F with braided' := fun X Y => by rw [F.braided]; simp }
 #align category_theory.braided_functor.to_lax_braided_functor CategoryTheory.BraidedFunctor.toLaxBraidedFunctor
 
 /-- The identity braided monoidal functor. -/
@@ -509,9 +490,7 @@ def tensorμ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor
 theorem tensorμ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
     tensorμ C (X₁, X₂) (Y₁, Y₂) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) =
       (α_ X₁ X₂ (Y₁ ⊗ Y₂)).Hom ≫ (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ≫ (𝟙 X₁ ⊗ (β_ X₂ Y₁).Hom ⊗ 𝟙 Y₂) :=
-  by
-  dsimp [tensor_μ]
-  simp
+  by dsimp [tensor_μ]; simp
 #align category_theory.tensor_μ_def₁ CategoryTheory.tensorμ_def₁
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -523,9 +502,7 @@ theorem tensorμ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
 theorem tensorμ_def₂ (X₁ X₂ Y₁ Y₂ : C) :
     (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).Hom) ≫ (α_ X₁ X₂ (Y₁ ⊗ Y₂)).inv ≫ tensorμ C (X₁, X₂) (Y₁, Y₂) =
       (𝟙 X₁ ⊗ (β_ X₂ Y₁).Hom ⊗ 𝟙 Y₂) ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).Hom) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv :=
-  by
-  dsimp [tensor_μ]
-  simp
+  by dsimp [tensor_μ]; simp
 #align category_theory.tensor_μ_def₂ CategoryTheory.tensorμ_def₂
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/
@@ -856,9 +833,7 @@ def tensorMonoidal : MonoidalFunctor (C × C) C :=
     associativity' := fun X Y Z => tensor_associativity C X.1 X.2 Y.1 Y.2 Z.1 Z.2
     left_unitality' := fun ⟨X₁, X₂⟩ => tensor_left_unitality C X₁ X₂
     right_unitality' := fun ⟨X₁, X₂⟩ => tensor_right_unitality C X₁ X₂
-    μ_isIso := by
-      dsimp [tensor_μ]
-      infer_instance }
+    μ_isIso := by dsimp [tensor_μ]; infer_instance }
 #align category_theory.tensor_monoidal CategoryTheory.tensorMonoidal
 
 /- ./././Mathport/Syntax/Translate/Expr.lean:177:8: unsupported: ambiguous notation -/

mathlib commit https://github.com/leanprover-community/mathlib/commit/75e7fca56381d056096ce5d05e938f63a6567828

@@ -75,13 +75,13 @@ class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] whe
 
 restate_axiom braided_category.braiding_naturality'
 
-attribute [simp, reassoc.1] braided_category.braiding_naturality
+attribute [simp, reassoc] braided_category.braiding_naturality
 
 restate_axiom braided_category.hexagon_forward'
 
 restate_axiom braided_category.hexagon_reverse'
 
-attribute [reassoc.1] braided_category.hexagon_forward braided_category.hexagon_reverse
+attribute [reassoc] braided_category.hexagon_forward braided_category.hexagon_reverse
 
 open Category
 
@@ -328,7 +328,7 @@ class SymmetricCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] e
 
 restate_axiom symmetric_category.symmetry'
 
-attribute [simp, reassoc.1] symmetric_category.symmetry
+attribute [simp, reassoc] symmetric_category.symmetry
 
 initialize_simps_projections SymmetricCategory (to_braided_category_braiding → braiding,
   -toBraidedCategory)

mathlib commit https://github.com/leanprover-community/mathlib/commit/284fdd2962e67d2932fa3a79ce19fcf92d38e228

@@ -4,7 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
 
 ! This file was ported from Lean 3 source module category_theory.monoidal.braided
-! leanprover-community/mathlib commit c9c9fa15fec7ca18e9ec97306fb8764bfe988a7e
+! leanprover-community/mathlib commit 2efd2423f8d25fa57cf7a179f5d8652ab4d0df44
 ! Please do not edit these lines, except to modify the commit id
 ! if you have ported upstream changes.
 -/
@@ -330,6 +330,9 @@ restate_axiom symmetric_category.symmetry'
 
 attribute [simp, reassoc.1] symmetric_category.symmetry
 
+initialize_simps_projections SymmetricCategory (to_braided_category_braiding → braiding,
+  -toBraidedCategory)
+
 variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory C] [BraidedCategory C]
 
 variable (D : Type u₂) [Category.{v₂} D] [MonoidalCategory D] [BraidedCategory D]

mathlib commit https://github.com/leanprover-community/mathlib/commit/bd9851ca476957ea4549eb19b40e7b5ade9428cc

These notions on functors are now Functor.Full, Functor.Faithful, Functor.EssSurj, Functor.IsEquivalence, Functor.ReflectsIsomorphisms. Deprecated aliases are introduced for the previous names.

@@ -196,7 +196,7 @@ Verifying the axioms for a braiding by checking that the candidate braiding is s
 by a faithful monoidal functor.
 -/
 def braidedCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalCategory C]
-    [MonoidalCategory D] (F : MonoidalFunctor C D) [Faithful F.toFunctor] [BraidedCategory D]
+    [MonoidalCategory D] (F : MonoidalFunctor C D) [F.Faithful] [BraidedCategory D]
     (β : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X)
     (w : ∀ X Y, F.μ _ _ ≫ F.map (β X Y).hom = (β_ _ _).hom ≫ F.μ _ _) : BraidedCategory C where
   braiding := β
@@ -239,8 +239,8 @@ def braidedCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalC
 
 /-- Pull back a braiding along a fully faithful monoidal functor. -/
 noncomputable def braidedCategoryOfFullyFaithful {C D : Type*} [Category C] [Category D]
-    [MonoidalCategory C] [MonoidalCategory D] (F : MonoidalFunctor C D) [Full F.toFunctor]
-    [Faithful F.toFunctor] [BraidedCategory D] : BraidedCategory C :=
+    [MonoidalCategory C] [MonoidalCategory D] (F : MonoidalFunctor C D) [F.Full]
+    [F.Faithful] [BraidedCategory D] : BraidedCategory C :=
   braidedCategoryOfFaithful F
     (fun X Y => F.toFunctor.preimageIso
       ((asIso (F.μ _ _)).symm ≪≫ β_ (F.obj X) (F.obj Y) ≪≫ asIso (F.μ _ _)))
@@ -446,7 +446,7 @@ A braided category with a faithful braided functor to a symmetric category is it
 -/
 def symmetricCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalCategory C]
     [MonoidalCategory D] [BraidedCategory C] [SymmetricCategory D] (F : BraidedFunctor C D)
-    [Faithful F.toFunctor] : SymmetricCategory C where
+    [F.Faithful] : SymmetricCategory C where
   symmetry X Y := F.map_injective (by simp)
 #align category_theory.symmetric_category_of_faithful CategoryTheory.symmetricCategoryOfFaithful
 

More tactics that are not used, found using the linter at #11308.

The PR consists of tactic removals, whitespace changes and replacing a porting note by an explanation.

@@ -94,7 +94,6 @@ theorem braiding_tensor_left (X Y Z : C) :
     (β_ (X ⊗ Y) Z).hom  =
       (α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫
         (β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom := by
-  intros
   apply (cancel_epi (α_ X Y Z).inv).1
   apply (cancel_mono (α_ Z X Y).inv).1
   simp [hexagon_reverse]
@@ -104,7 +103,6 @@ theorem braiding_tensor_right (X Y Z : C) :
     (β_ X (Y ⊗ Z)).hom  =
       (α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
         Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv := by
-  intros
   apply (cancel_epi (α_ X Y Z).hom).1
   apply (cancel_mono (α_ Y Z X).hom).1
   simp [hexagon_forward]

Extracted from #6307

@@ -291,7 +291,6 @@ theorem braiding_leftUnitor_aux₂ (X : C) :
       by (slice_lhs 2 3 => rw [← braiding_naturality_right]); simp only [assoc]
     _ = (α_ _ _ _).hom ≫ (_ ◁ (λ_ _).hom) := by rw [Iso.hom_inv_id, comp_id]
     _ = (ρ_ X).hom ▷ 𝟙_ C := by rw [triangle]
-
 #align category_theory.braiding_left_unitor_aux₂ CategoryTheory.braiding_leftUnitor_aux₂
 
 @[reassoc]
@@ -324,7 +323,6 @@ theorem braiding_rightUnitor_aux₂ (X : C) :
       by (slice_lhs 2 3 => rw [← braiding_naturality_left]); simp only [assoc]
     _ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ▷ _) := by rw [Iso.hom_inv_id, comp_id]
     _ = 𝟙_ C ◁ (λ_ X).hom := by rw [triangle_assoc_comp_right]
-
 #align category_theory.braiding_right_unitor_aux₂ CategoryTheory.braiding_rightUnitor_aux₂
 
 @[reassoc]

We set id_tensorHom and tensorHom_id as simp lemmas. Partially extracted from #6307.

@@ -206,37 +206,37 @@ def braidedCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalC
     intros
     apply F.map_injective
     refine (cancel_epi (F.μ ?_ ?_)).1 ?_
-    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_left'_assoc, w, Functor.map_comp,
-      reassoc_of% w, braiding_naturality_left_assoc, LaxMonoidalFunctor.μ_natural_right']
+    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_left_assoc, w, Functor.map_comp,
+      reassoc_of% w, braiding_naturality_left_assoc, LaxMonoidalFunctor.μ_natural_right]
   braiding_naturality_right := by
     intros
     apply F.map_injective
     refine (cancel_epi (F.μ ?_ ?_)).1 ?_
-    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_right'_assoc, w, Functor.map_comp,
-      reassoc_of% w, braiding_naturality_right_assoc, LaxMonoidalFunctor.μ_natural_left']
+    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_right_assoc, w, Functor.map_comp,
+      reassoc_of% w, braiding_naturality_right_assoc, LaxMonoidalFunctor.μ_natural_left]
   hexagon_forward := by
     intros
     apply F.map_injective
     refine (cancel_epi (F.μ _ _)).1 ?_
     refine (cancel_epi (F.μ _ _ ▷ _)).1 ?_
     rw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, ←
-      LaxMonoidalFunctor.μ_natural_left'_assoc, ← comp_whiskerRight_assoc, w,
-      comp_whiskerRight_assoc, LaxMonoidalFunctor.associativity'_assoc,
-      LaxMonoidalFunctor.associativity'_assoc, ← LaxMonoidalFunctor.μ_natural_right', ←
+      LaxMonoidalFunctor.μ_natural_left_assoc, ← comp_whiskerRight_assoc, w,
+      comp_whiskerRight_assoc, LaxMonoidalFunctor.associativity_assoc,
+      LaxMonoidalFunctor.associativity_assoc, ← LaxMonoidalFunctor.μ_natural_right, ←
       MonoidalCategory.whiskerLeft_comp_assoc, w, MonoidalCategory.whiskerLeft_comp_assoc,
       reassoc_of% w, braiding_naturality_right_assoc,
-      LaxMonoidalFunctor.associativity', hexagon_forward_assoc]
+      LaxMonoidalFunctor.associativity, hexagon_forward_assoc]
   hexagon_reverse := by
     intros
     apply F.toFunctor.map_injective
     refine (cancel_epi (F.μ _ _)).1 ?_
     refine (cancel_epi (_ ◁ F.μ _ _)).1 ?_
     rw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, ←
-      LaxMonoidalFunctor.μ_natural_right'_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc, w,
-      MonoidalCategory.whiskerLeft_comp_assoc, LaxMonoidalFunctor.associativity_inv'_assoc,
-      LaxMonoidalFunctor.associativity_inv'_assoc, ← LaxMonoidalFunctor.μ_natural_left',
+      LaxMonoidalFunctor.μ_natural_right_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc, w,
+      MonoidalCategory.whiskerLeft_comp_assoc, LaxMonoidalFunctor.associativity_inv_assoc,
+      LaxMonoidalFunctor.associativity_inv_assoc, ← LaxMonoidalFunctor.μ_natural_left,
       ← comp_whiskerRight_assoc, w, comp_whiskerRight_assoc, reassoc_of% w,
-      braiding_naturality_left_assoc, LaxMonoidalFunctor.associativity_inv', hexagon_reverse_assoc]
+      braiding_naturality_left_assoc, LaxMonoidalFunctor.associativity_inv, hexagon_reverse_assoc]
 #align category_theory.braided_category_of_faithful CategoryTheory.braidedCategoryOfFaithful
 
 /-- Pull back a braiding along a fully faithful monoidal functor. -/
@@ -290,7 +290,7 @@ theorem braiding_leftUnitor_aux₂ (X : C) :
     _ = (α_ _ _ _).hom ≫ (_ ◁ (λ_ _).hom) ≫ (β_ _ _).hom ≫ (β_ X _).inv :=
       by (slice_lhs 2 3 => rw [← braiding_naturality_right]); simp only [assoc]
     _ = (α_ _ _ _).hom ≫ (_ ◁ (λ_ _).hom) := by rw [Iso.hom_inv_id, comp_id]
-    _ = (ρ_ X).hom ▷ 𝟙_ C := by rw [triangle']
+    _ = (ρ_ X).hom ▷ 𝟙_ C := by rw [triangle]
 
 #align category_theory.braiding_left_unitor_aux₂ CategoryTheory.braiding_leftUnitor_aux₂
 
@@ -323,7 +323,7 @@ theorem braiding_rightUnitor_aux₂ (X : C) :
     _ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ▷ _) ≫ (β_ _ X).hom ≫ (β_ _ _).inv :=
       by (slice_lhs 2 3 => rw [← braiding_naturality_left]); simp only [assoc]
     _ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ▷ _) := by rw [Iso.hom_inv_id, comp_id]
-    _ = 𝟙_ C ◁ (λ_ X).hom := by rw [triangle_assoc_comp_right']
+    _ = 𝟙_ C ◁ (λ_ X).hom := by rw [triangle_assoc_comp_right]
 
 #align category_theory.braiding_right_unitor_aux₂ CategoryTheory.braiding_rightUnitor_aux₂
 
@@ -552,8 +552,6 @@ theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ :
   simp only [assoc]
 #align category_theory.tensor_μ_natural CategoryTheory.tensor_μ_natural
 
-attribute [local simp] id_tensorHom tensorHom_id
-
 @[reassoc]
 theorem tensor_μ_natural_left {X₁ X₂ Y₁ Y₂ : C} (f₁: X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (Z₁ Z₂ : C) :
     (f₁ ⊗ f₂) ▷ (Z₁ ⊗ Z₂) ≫ tensor_μ C (Y₁, Y₂) (Z₁, Z₂) =
@@ -737,11 +735,10 @@ monoidal opposite, upgraded to a braided functor. -/
 @[simps!] def mopBraidedFunctor : BraidedFunctor C Cᴹᵒᵖ where
   μ X Y := (β_ (mop X) (mop Y)).hom
   ε := 𝟙 (𝟙_ Cᴹᵒᵖ)
-  -- `id_tensorHom`, `tensorHom_id` should be simp lemmas when #6307 is merged
-  -- we could then make this fully automated if we mark `yang_baxter` as simp
+  -- we could make this fully automated if we mark `← yang_baxter_assoc` as simp
   -- should it be marked as such?
   associativity X Y Z := by
-    simp [id_tensorHom, tensorHom_id, ← yang_baxter_assoc]
+    simp [← yang_baxter_assoc]
   __ := mopFunctor C
 
 /-- The identity functor on `C`, viewed as a functor from the
@@ -750,7 +747,7 @@ monoidal opposite of `C` to `C`, upgraded to a braided functor. -/
   μ X Y := (β_ (unmop X) (unmop Y)).hom
   ε := 𝟙 (𝟙_ C)
   associativity X Y Z := by
-    simp [id_tensorHom, tensorHom_id, ← yang_baxter_assoc]
+    simp [← yang_baxter_assoc]
   __ := unmopFunctor C
 
 end MonoidalOpposite

This PR adds some basics about monoidal opposite categories and their relation to the original category, as well as the Yang-Baxter equation for braided monoidal categories. It should be easy to define an action of the braid group on an object of a braided monoidal category from this.

@@ -3,10 +3,12 @@ Copyright (c) 2020 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
 -/
-import Mathlib.CategoryTheory.Monoidal.Discrete
 import Mathlib.CategoryTheory.Monoidal.Free.Coherence
+import Mathlib.CategoryTheory.Monoidal.Discrete
 import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
+import Mathlib.CategoryTheory.Monoidal.Opposite
 import Mathlib.Tactic.CategoryTheory.Coherence
+import Mathlib.CategoryTheory.CommSq
 
 #align_import category_theory.monoidal.braided from "leanprover-community/mathlib"@"2efd2423f8d25fa57cf7a179f5d8652ab4d0df44"
 
@@ -26,6 +28,10 @@ The rationale is that we are not carrying any additional data, just requiring a
 * Construct the Drinfeld center of a monoidal category as a braided monoidal category.
 * Say something about pseudo-natural transformations.
 
+## References
+
+* [Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, Victor Ostrik, *Tensor categories*][egno15]
+
 -/
 
 
@@ -83,7 +89,7 @@ namespace BraidedCategory
 
 variable {C : Type u} [Category.{v} C] [MonoidalCategory.{v} C] [BraidedCategory.{v} C]
 
-@[simp]
+@[simp, reassoc]
 theorem braiding_tensor_left (X Y Z : C) :
     (β_ (X ⊗ Y) Z).hom  =
       (α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫
@@ -93,7 +99,7 @@ theorem braiding_tensor_left (X Y Z : C) :
   apply (cancel_mono (α_ Z X Y).inv).1
   simp [hexagon_reverse]
 
-@[simp]
+@[simp, reassoc]
 theorem braiding_tensor_right (X Y Z : C) :
     (β_ X (Y ⊗ Z)).hom  =
       (α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
@@ -103,14 +109,14 @@ theorem braiding_tensor_right (X Y Z : C) :
   apply (cancel_mono (α_ Y Z X).hom).1
   simp [hexagon_forward]
 
-@[simp]
+@[simp, reassoc]
 theorem braiding_inv_tensor_left (X Y Z : C) :
     (β_ (X ⊗ Y) Z).inv  =
       (α_ Z X Y).inv ≫ (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫
         X ◁ (β_ Y Z).inv ≫ (α_ X Y Z).inv :=
   eq_of_inv_eq_inv (by simp)
 
-@[simp]
+@[simp, reassoc]
 theorem braiding_inv_tensor_right (X Y Z : C) :
     (β_ X (Y ⊗ Z)).inv  =
       (α_ Y Z X).hom ≫ Y ◁ (β_ X Z).inv ≫ (α_ Y X Z).inv ≫
@@ -123,6 +129,68 @@ theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
   rw [tensorHom_def' f g, tensorHom_def g f]
   simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc]
 
+@[reassoc (attr := simp)]
+theorem braiding_inv_naturality_right (X : C) {Y Z : C} (f : Y ⟶ Z) :
+    X ◁ f ≫ (β_ Z X).inv = (β_ Y X).inv ≫ f ▷ X :=
+  CommSq.w <| .vert_inv <| .mk <| braiding_naturality_left f X
+
+@[reassoc (attr := simp)]
+theorem braiding_inv_naturality_left {X Y : C} (f : X ⟶ Y) (Z : C) :
+    f ▷ Z ≫ (β_ Z Y).inv = (β_ Z X).inv ≫ Z ◁ f :=
+  CommSq.w <| .vert_inv <| .mk <| braiding_naturality_right Z f
+
+@[reassoc (attr := simp)]
+theorem braiding_inv_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
+    (f ⊗ g) ≫ (β_ Y' Y).inv = (β_ X' X).inv ≫ (g ⊗ f) :=
+  CommSq.w <| .vert_inv <| .mk <| braiding_naturality g f
+
+@[reassoc]
+theorem yang_baxter (X Y Z : C) :
+    (α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
+    Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv ≫ (β_ Y Z).hom ▷ X ≫ (α_ Z Y X).hom =
+      X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫ (β_ X Z).hom ▷ Y ≫
+      (α_ Z X Y).hom ≫ Z ◁ (β_ X Y).hom := by
+  rw [← braiding_tensor_right_assoc X Y Z, ← cancel_mono (α_ Z Y X).inv]
+  repeat rw [assoc]
+  rw [Iso.hom_inv_id, comp_id, ← braiding_naturality_right, braiding_tensor_right]
+
+theorem yang_baxter' (X Y Z : C) :
+    (β_ X Y).hom ▷ Z ⊗≫ Y ◁ (β_ X Z).hom ⊗≫ (β_ Y Z).hom ▷ X =
+      𝟙 _ ⊗≫ (X ◁ (β_ Y Z).hom ⊗≫ (β_ X Z).hom ▷ Y ⊗≫ Z ◁ (β_ X Y).hom) ⊗≫ 𝟙 _ := by
+  rw [← cancel_epi (α_ X Y Z).inv, ← cancel_mono (α_ Z Y X).hom]
+  convert yang_baxter X Y Z using 1
+  all_goals coherence
+
+theorem yang_baxter_iso (X Y Z : C) :
+    (α_ X Y Z).symm ≪≫ whiskerRightIso (β_ X Y) Z ≪≫ α_ Y X Z ≪≫
+    whiskerLeftIso Y (β_ X Z) ≪≫ (α_ Y Z X).symm ≪≫
+    whiskerRightIso (β_ Y Z) X ≪≫ (α_ Z Y X) =
+      whiskerLeftIso X (β_ Y Z) ≪≫ (α_ X Z Y).symm ≪≫
+      whiskerRightIso (β_ X Z) Y ≪≫ α_ Z X Y ≪≫
+      whiskerLeftIso Z (β_ X Y) := Iso.ext (yang_baxter X Y Z)
+
+theorem hexagon_forward_iso (X Y Z : C) :
+    α_ X Y Z ≪≫ β_ X (Y ⊗ Z) ≪≫ α_ Y Z X =
+      whiskerRightIso (β_ X Y) Z ≪≫ α_ Y X Z ≪≫ whiskerLeftIso Y (β_ X Z) :=
+  Iso.ext (hexagon_forward X Y Z)
+
+theorem hexagon_reverse_iso (X Y Z : C) :
+    (α_ X Y Z).symm ≪≫ β_ (X ⊗ Y) Z ≪≫ (α_ Z X Y).symm =
+      whiskerLeftIso X (β_ Y Z) ≪≫ (α_ X Z Y).symm ≪≫ whiskerRightIso (β_ X Z) Y :=
+  Iso.ext (hexagon_reverse X Y Z)
+
+@[reassoc]
+theorem hexagon_forward_inv (X Y Z : C) :
+    (α_ Y Z X).inv ≫ (β_ X (Y ⊗ Z)).inv ≫ (α_ X Y Z).inv =
+      Y ◁ (β_ X Z).inv ≫ (α_ Y X Z).inv ≫ (β_ X Y).inv ▷ Z := by
+  simp
+
+@[reassoc]
+theorem hexagon_reverse_inv (X Y Z : C) :
+    (α_ Z X Y).hom ≫ (β_ (X ⊗ Y) Z).inv ≫ (α_ X Y Z).hom =
+      (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫ X ◁ (β_ Y Z).inv := by
+  simp
+
 end BraidedCategory
 
 /--
@@ -298,6 +366,10 @@ class SymmetricCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] e
 
 attribute [reassoc (attr := simp)] SymmetricCategory.symmetry
 
+lemma SymmetricCategory.braiding_swap_eq_inv_braiding {C : Type u₁}
+    [Category.{v₁} C] [MonoidalCategory C] [SymmetricCategory C] (X Y : C) :
+    (β_ Y X).hom = (β_ X Y).inv := Iso.inv_ext' (symmetry X Y)
+
 variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory C] [BraidedCategory C]
 variable (D : Type u₂) [Category.{v₂} D] [MonoidalCategory D] [BraidedCategory D]
 variable (E : Type u₃) [Category.{v₃} E] [MonoidalCategory E] [BraidedCategory E]
@@ -618,4 +690,93 @@ attribute [reassoc] associator_monoidal
 
 end Tensor
 
+instance : BraidedCategory Cᵒᵖ where
+  braiding X Y := (β_ Y.unop X.unop).op
+  braiding_naturality_right X {_ _} f := Quiver.Hom.unop_inj <| by simp
+  braiding_naturality_left {_ _} f Z := Quiver.Hom.unop_inj <| by simp
+
+section OppositeLemmas
+
+open Opposite
+
+@[simp] lemma op_braiding (X Y : C) : (β_ X Y).op = β_ (op Y) (op X) := rfl
+@[simp] lemma unop_braiding (X Y : Cᵒᵖ) : (β_ X Y).unop = β_ (unop Y) (unop X) := rfl
+
+@[simp] lemma op_hom_braiding (X Y : C) : (β_ X Y).hom.op = (β_ (op Y) (op X)).hom := rfl
+@[simp] lemma unop_hom_braiding (X Y : Cᵒᵖ) : (β_ X Y).hom.unop = (β_ (unop Y) (unop X)).hom := rfl
+
+@[simp] lemma op_inv_braiding (X Y : C) : (β_ X Y).inv.op = (β_ (op Y) (op X)).inv := rfl
+@[simp] lemma unop_inv_braiding (X Y : Cᵒᵖ) : (β_ X Y).inv.unop = (β_ (unop Y) (unop X)).inv := rfl
+
+end OppositeLemmas
+
+namespace MonoidalOpposite
+
+instance instBraiding : BraidedCategory Cᴹᵒᵖ where
+  braiding X Y := (β_ Y.unmop X.unmop).mop
+  braiding_naturality_right X {_ _} f := Quiver.Hom.unmop_inj <| by simp
+  braiding_naturality_left {_ _} f Z := Quiver.Hom.unmop_inj <| by simp
+
+section MonoidalOppositeLemmas
+
+@[simp] lemma mop_braiding (X Y : C) : (β_ X Y).mop = β_ (mop Y) (mop X) := rfl
+@[simp] lemma unmop_braiding (X Y : Cᴹᵒᵖ) : (β_ X Y).unmop = β_ (unmop Y) (unmop X) := rfl
+
+@[simp] lemma mop_hom_braiding (X Y : C) : (β_ X Y).hom.mop = (β_ (mop Y) (mop X)).hom := rfl
+@[simp]
+lemma unmop_hom_braiding (X Y : Cᴹᵒᵖ) : (β_ X Y).hom.unmop = (β_ (unmop Y) (unmop X)).hom := rfl
+
+@[simp] lemma mop_inv_braiding (X Y : C) : (β_ X Y).inv.mop = (β_ (mop Y) (mop X)).inv := rfl
+@[simp]
+lemma unmop_inv_braiding (X Y : Cᴹᵒᵖ) : (β_ X Y).inv.unmop = (β_ (unmop Y) (unmop X)).inv := rfl
+
+end MonoidalOppositeLemmas
+
+/-- The identity functor on `C`, viewed as a functor from `C` to its
+monoidal opposite, upgraded to a braided functor. -/
+@[simps!] def mopBraidedFunctor : BraidedFunctor C Cᴹᵒᵖ where
+  μ X Y := (β_ (mop X) (mop Y)).hom
+  ε := 𝟙 (𝟙_ Cᴹᵒᵖ)
+  -- `id_tensorHom`, `tensorHom_id` should be simp lemmas when #6307 is merged
+  -- we could then make this fully automated if we mark `yang_baxter` as simp
+  -- should it be marked as such?
+  associativity X Y Z := by
+    simp [id_tensorHom, tensorHom_id, ← yang_baxter_assoc]
+  __ := mopFunctor C
+
+/-- The identity functor on `C`, viewed as a functor from the
+monoidal opposite of `C` to `C`, upgraded to a braided functor. -/
+@[simps!] def unmopBraidedFunctor : BraidedFunctor Cᴹᵒᵖ C where
+  μ X Y := (β_ (unmop X) (unmop Y)).hom
+  ε := 𝟙 (𝟙_ C)
+  associativity X Y Z := by
+    simp [id_tensorHom, tensorHom_id, ← yang_baxter_assoc]
+  __ := unmopFunctor C
+
+end MonoidalOpposite
+
+/-- The braided monoidal category obtained from `C` by replacing its braiding
+`β_ X Y : X ⊗ Y ≅ Y ⊗ X` with the inverse `(β_ Y X)⁻¹ : X ⊗ Y ≅ Y ⊗ X`.
+This corresponds to the automorphism of the braid group swapping
+over-crossings and under-crossings. -/
+@[reducible] def reverseBraiding : BraidedCategory C where
+  braiding X Y := (β_ Y X).symm
+  braiding_naturality_right X {_ _} f := by simp
+  braiding_naturality_left {_ _} f Z := by simp
+
+lemma SymmetricCategory.reverseBraiding_eq (C : Type u₁) [Category.{v₁} C]
+    [MonoidalCategory C] [i : SymmetricCategory C] :
+    reverseBraiding C = i.toBraidedCategory := by
+  dsimp only [reverseBraiding]
+  congr
+  funext X Y
+  exact Iso.ext (braiding_swap_eq_inv_braiding Y X).symm
+
+/-- The identity functor from `C` to `C`, where the codomain is given the
+reversed braiding, upgraded to a braided functor. -/
+def SymmetricCategory.equivReverseBraiding (C : Type u₁) [Category.{v₁} C]
+    [MonoidalCategory C] [SymmetricCategory C] :=
+  @BraidedFunctor.mk C _ _ _ C _ _ (reverseBraiding C) (.id C) <| by
+    intros; simp [reverseBraiding, braiding_swap_eq_inv_braiding]
+
 end CategoryTheory

• depends on: #9830
• depends on: #9772
• depends on: #9996

This checks files for unused imports. The output here is piped through gh-problem-matcher-wrap so that it will show up as annotations.

Co-authored-by: Mario Carneiro <di.gama@gmail.com>

@@ -3,9 +3,10 @@ Copyright (c) 2020 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
 -/
-import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
-import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
 import Mathlib.CategoryTheory.Monoidal.Discrete
+import Mathlib.CategoryTheory.Monoidal.Free.Coherence
+import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
+import Mathlib.Tactic.CategoryTheory.Coherence
 
 #align_import category_theory.monoidal.braided from "leanprover-community/mathlib"@"2efd2423f8d25fa57cf7a179f5d8652ab4d0df44"
 

• depends on: #10061

@@ -42,26 +42,25 @@ and also satisfies the two hexagon identities.
 class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] where
   /-- The braiding natural isomorphism. -/
   braiding : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X
-  -- Note: `𝟙 X ⊗ f` will be replaced by `X ◁ f` (and similarly for `f ⊗ 𝟙 Z`) in #6307.
   braiding_naturality_right :
     ∀ (X : C) {Y Z : C} (f : Y ⟶ Z),
-      (𝟙 X ⊗ f) ≫ (braiding X Z).hom = (braiding X Y).hom ≫ (f ⊗ 𝟙 X) := by
+      X ◁ f ≫ (braiding X Z).hom = (braiding X Y).hom ≫ f ▷ X := by
     aesop_cat
   braiding_naturality_left :
     ∀ {X Y : C} (f : X ⟶ Y) (Z : C),
-      (f ⊗ 𝟙 Z) ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ (𝟙 Z ⊗ f) := by
+      f ▷ Z ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ Z ◁ f := by
     aesop_cat
   /-- The first hexagon identity. -/
   hexagon_forward :
     ∀ X Y Z : C,
       (α_ X Y Z).hom ≫ (braiding X (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom =
-        ((braiding X Y).hom ⊗ 𝟙 Z) ≫ (α_ Y X Z).hom ≫ (𝟙 Y ⊗ (braiding X Z).hom) := by
+        ((braiding X Y).hom ▷ Z) ≫ (α_ Y X Z).hom ≫ (Y ◁ (braiding X Z).hom) := by
     aesop_cat
   /-- The second hexagon identity. -/
   hexagon_reverse :
     ∀ X Y Z : C,
       (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =
-        (𝟙 X ⊗ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ⊗ 𝟙 Y) := by
+        (X ◁ (braiding Y Z).hom) ≫ (α_ X Z Y).inv ≫ ((braiding X Z).hom ▷ Y) := by
     aesop_cat
 #align category_theory.braided_category CategoryTheory.BraidedCategory
 
@@ -86,8 +85,8 @@ variable {C : Type u} [Category.{v} C] [MonoidalCategory.{v} C] [BraidedCategory
 @[simp]
 theorem braiding_tensor_left (X Y Z : C) :
     (β_ (X ⊗ Y) Z).hom  =
-      (α_ X Y Z).hom ≫ (𝟙 X ⊗ (β_ Y Z).hom) ≫ (α_ X Z Y).inv ≫
-        ((β_ X Z).hom ⊗ 𝟙 Y) ≫ (α_ Z X Y).hom := by
+      (α_ X Y Z).hom ≫ X ◁ (β_ Y Z).hom ≫ (α_ X Z Y).inv ≫
+        (β_ X Z).hom ▷ Y ≫ (α_ Z X Y).hom := by
   intros
   apply (cancel_epi (α_ X Y Z).inv).1
   apply (cancel_mono (α_ Z X Y).inv).1
@@ -96,8 +95,8 @@ theorem braiding_tensor_left (X Y Z : C) :
 @[simp]
 theorem braiding_tensor_right (X Y Z : C) :
     (β_ X (Y ⊗ Z)).hom  =
-      (α_ X Y Z).inv ≫ ((β_ X Y).hom ⊗ 𝟙 Z) ≫ (α_ Y X Z).hom ≫
-        (𝟙 Y ⊗ (β_ X Z).hom) ≫ (α_ Y Z X).inv := by
+      (α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
+        Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv := by
   intros
   apply (cancel_epi (α_ X Y Z).hom).1
   apply (cancel_mono (α_ Y Z X).hom).1
@@ -106,22 +105,21 @@ theorem braiding_tensor_right (X Y Z : C) :
 @[simp]
 theorem braiding_inv_tensor_left (X Y Z : C) :
     (β_ (X ⊗ Y) Z).inv  =
-      (α_ Z X Y).inv ≫ ((β_ X Z).inv ⊗ 𝟙 Y) ≫ (α_ X Z Y).hom ≫
-        (𝟙 X ⊗ (β_ Y Z).inv) ≫ (α_ X Y Z).inv :=
+      (α_ Z X Y).inv ≫ (β_ X Z).inv ▷ Y ≫ (α_ X Z Y).hom ≫
+        X ◁ (β_ Y Z).inv ≫ (α_ X Y Z).inv :=
   eq_of_inv_eq_inv (by simp)
 
 @[simp]
 theorem braiding_inv_tensor_right (X Y Z : C) :
     (β_ X (Y ⊗ Z)).inv  =
-      (α_ Y Z X).hom ≫ (𝟙 Y ⊗ (β_ X Z).inv) ≫ (α_ Y X Z).inv ≫
-        ((β_ X Y).inv ⊗ 𝟙 Z) ≫ (α_ X Y Z).hom :=
+      (α_ Y Z X).hom ≫ Y ◁ (β_ X Z).inv ≫ (α_ Y X Z).inv ≫
+        (β_ X Y).inv ▷ Z ≫ (α_ X Y Z).hom :=
   eq_of_inv_eq_inv (by simp)
 
--- The priority setting will not be needed when we replace `𝟙 X ⊗ f` by `X ◁ f`.
-@[reassoc (attr := simp (low))]
+@[reassoc (attr := simp)]
 theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
     (f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) := by
-  rw [← tensor_id_comp_id_tensor f g, ← id_tensor_comp_tensor_id g f]
+  rw [tensorHom_def' f g, tensorHom_def g f]
   simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc]
 
 end BraidedCategory
@@ -139,36 +137,37 @@ def braidedCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalC
     intros
     apply F.map_injective
     refine (cancel_epi (F.μ ?_ ?_)).1 ?_
-    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_left_assoc, w, Functor.map_comp,
-      reassoc_of% w, braiding_naturality_left_assoc, LaxMonoidalFunctor.μ_natural_right]
+    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_left'_assoc, w, Functor.map_comp,
+      reassoc_of% w, braiding_naturality_left_assoc, LaxMonoidalFunctor.μ_natural_right']
   braiding_naturality_right := by
     intros
     apply F.map_injective
     refine (cancel_epi (F.μ ?_ ?_)).1 ?_
-    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_right_assoc, w, Functor.map_comp,
-      reassoc_of% w, braiding_naturality_right_assoc, LaxMonoidalFunctor.μ_natural_left]
+    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_right'_assoc, w, Functor.map_comp,
+      reassoc_of% w, braiding_naturality_right_assoc, LaxMonoidalFunctor.μ_natural_left']
   hexagon_forward := by
     intros
     apply F.map_injective
     refine (cancel_epi (F.μ _ _)).1 ?_
-    refine (cancel_epi (F.μ _ _ ⊗ 𝟙 _)).1 ?_
+    refine (cancel_epi (F.μ _ _ ▷ _)).1 ?_
     rw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, ←
-      LaxMonoidalFunctor.μ_natural_assoc, Functor.map_id, ← comp_tensor_id_assoc, w,
-      comp_tensor_id, assoc, LaxMonoidalFunctor.associativity_assoc,
-      LaxMonoidalFunctor.associativity_assoc, ← LaxMonoidalFunctor.μ_natural, Functor.map_id, ←
-      id_tensor_comp_assoc, w, id_tensor_comp_assoc, reassoc_of% w, braiding_naturality_assoc,
-      LaxMonoidalFunctor.associativity, hexagon_forward_assoc]
+      LaxMonoidalFunctor.μ_natural_left'_assoc, ← comp_whiskerRight_assoc, w,
+      comp_whiskerRight_assoc, LaxMonoidalFunctor.associativity'_assoc,
+      LaxMonoidalFunctor.associativity'_assoc, ← LaxMonoidalFunctor.μ_natural_right', ←
+      MonoidalCategory.whiskerLeft_comp_assoc, w, MonoidalCategory.whiskerLeft_comp_assoc,
+      reassoc_of% w, braiding_naturality_right_assoc,
+      LaxMonoidalFunctor.associativity', hexagon_forward_assoc]
   hexagon_reverse := by
     intros
     apply F.toFunctor.map_injective
     refine (cancel_epi (F.μ _ _)).1 ?_
-    refine (cancel_epi (𝟙 _ ⊗ F.μ _ _)).1 ?_
+    refine (cancel_epi (_ ◁ F.μ _ _)).1 ?_
     rw [Functor.map_comp, Functor.map_comp, Functor.map_comp, Functor.map_comp, ←
-      LaxMonoidalFunctor.μ_natural_assoc, Functor.map_id, ← id_tensor_comp_assoc, w,
-      id_tensor_comp_assoc, LaxMonoidalFunctor.associativity_inv_assoc,
-      LaxMonoidalFunctor.associativity_inv_assoc, ← LaxMonoidalFunctor.μ_natural,
-      Functor.map_id, ← comp_tensor_id_assoc, w, comp_tensor_id_assoc, reassoc_of% w,
-      braiding_naturality_assoc, LaxMonoidalFunctor.associativity_inv, hexagon_reverse_assoc]
+      LaxMonoidalFunctor.μ_natural_right'_assoc, ← MonoidalCategory.whiskerLeft_comp_assoc, w,
+      MonoidalCategory.whiskerLeft_comp_assoc, LaxMonoidalFunctor.associativity_inv'_assoc,
+      LaxMonoidalFunctor.associativity_inv'_assoc, ← LaxMonoidalFunctor.μ_natural_left',
+      ← comp_whiskerRight_assoc, w, comp_whiskerRight_assoc, reassoc_of% w,
+      braiding_naturality_left_assoc, LaxMonoidalFunctor.associativity_inv', hexagon_reverse_assoc]
 #align category_theory.braided_category_of_faithful CategoryTheory.braidedCategoryOfFaithful
 
 /-- Pull back a braiding along a fully faithful monoidal functor. -/
@@ -200,82 +199,89 @@ variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory C] [BraidedCatego
 
 theorem braiding_leftUnitor_aux₁ (X : C) :
     (α_ (𝟙_ C) (𝟙_ C) X).hom ≫
-        (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ⊗ 𝟙 _) =
-      ((λ_ _).hom ⊗ 𝟙 X) ≫ (β_ X (𝟙_ C)).inv :=
-  by rw [← leftUnitor_tensor, leftUnitor_naturality]; simp [id_tensorHom, tensorHom_id]
+        (𝟙_ C ◁ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ▷ _) =
+      ((λ_ _).hom ▷ X) ≫ (β_ X (𝟙_ C)).inv := by
+  coherence
 #align category_theory.braiding_left_unitor_aux₁ CategoryTheory.braiding_leftUnitor_aux₁
 
 theorem braiding_leftUnitor_aux₂ (X : C) :
-    ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) = (ρ_ X).hom ⊗ 𝟙 (𝟙_ C) :=
+    ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ ((λ_ X).hom ▷ 𝟙_ C) = (ρ_ X).hom ▷ 𝟙_ C :=
   calc
-    ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) =
-      ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (α_ _ _ _).hom ≫ (α_ _ _ _).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) :=
+    ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ ((λ_ X).hom ▷ 𝟙_ C) =
+      ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ (α_ _ _ _).hom ≫ (α_ _ _ _).inv ≫ ((λ_ X).hom ▷ 𝟙_ C) :=
       by coherence
-    _ = ((β_ X (𝟙_ C)).hom ⊗ 𝟙 (𝟙_ C)) ≫ (α_ _ _ _).hom ≫ (𝟙 _ ⊗ (β_ X _).hom) ≫
-          (𝟙 _ ⊗ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫ ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) :=
-      by (slice_rhs 3 4 => rw [← id_tensor_comp, Iso.hom_inv_id, tensor_id]); rw [id_comp]
-    _ = (α_ _ _ _).hom ≫ (β_ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 _ ⊗ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫
-          ((λ_ X).hom ⊗ 𝟙 (𝟙_ C)) :=
+    _ = ((β_ X (𝟙_ C)).hom ▷ 𝟙_ C) ≫ (α_ _ _ _).hom ≫ (_ ◁ (β_ X _).hom) ≫
+          (_ ◁ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫ ((λ_ X).hom ▷ 𝟙_ C) :=
+      by simp
+    _ = (α_ _ _ _).hom ≫ (β_ _ _).hom ≫ (α_ _ _ _).hom ≫ (_ ◁ (β_ X _).inv) ≫ (α_ _ _ _).inv ≫
+          ((λ_ X).hom ▷ 𝟙_ C) :=
       by (slice_lhs 1 3 => rw [← hexagon_forward]); simp only [assoc]
-    _ = (α_ _ _ _).hom ≫ (β_ _ _).hom ≫ ((λ_ _).hom ⊗ 𝟙 X) ≫ (β_ X _).inv :=
+    _ = (α_ _ _ _).hom ≫ (β_ _ _).hom ≫ ((λ_ _).hom ▷ X) ≫ (β_ X _).inv :=
       by rw [braiding_leftUnitor_aux₁]
-    _ = (α_ _ _ _).hom ≫ (𝟙 _ ⊗ (λ_ _).hom) ≫ (β_ _ _).hom ≫ (β_ X _).inv :=
-      by (slice_lhs 2 3 => rw [← braiding_naturality]); simp only [assoc]
-    _ = (α_ _ _ _).hom ≫ (𝟙 _ ⊗ (λ_ _).hom) := by rw [Iso.hom_inv_id, comp_id]
-    _ = (ρ_ X).hom ⊗ 𝟙 (𝟙_ C) := by rw [triangle]
+    _ = (α_ _ _ _).hom ≫ (_ ◁ (λ_ _).hom) ≫ (β_ _ _).hom ≫ (β_ X _).inv :=
+      by (slice_lhs 2 3 => rw [← braiding_naturality_right]); simp only [assoc]
+    _ = (α_ _ _ _).hom ≫ (_ ◁ (λ_ _).hom) := by rw [Iso.hom_inv_id, comp_id]
+    _ = (ρ_ X).hom ▷ 𝟙_ C := by rw [triangle']
 
 #align category_theory.braiding_left_unitor_aux₂ CategoryTheory.braiding_leftUnitor_aux₂
 
-@[simp]
+@[reassoc]
 theorem braiding_leftUnitor (X : C) : (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom = (ρ_ X).hom := by
-  rw [← tensor_right_iff, comp_tensor_id, braiding_leftUnitor_aux₂]
+  rw [← whiskerRight_iff, comp_whiskerRight, braiding_leftUnitor_aux₂]
 #align category_theory.braiding_left_unitor CategoryTheory.braiding_leftUnitor
 
 theorem braiding_rightUnitor_aux₁ (X : C) :
     (α_ X (𝟙_ C) (𝟙_ C)).inv ≫
-        ((β_ (𝟙_ C) X).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ _ X _).hom ≫ (𝟙 _ ⊗ (ρ_ X).hom) =
-      (𝟙 X ⊗ (ρ_ _).hom) ≫ (β_ (𝟙_ C) X).inv :=
-  by rw [← rightUnitor_tensor, rightUnitor_naturality]; simp [id_tensorHom, tensorHom_id]
+        ((β_ (𝟙_ C) X).inv ▷ 𝟙_ C) ≫ (α_ _ X _).hom ≫ (_ ◁ (ρ_ X).hom) =
+      (X ◁ (ρ_ _).hom) ≫ (β_ (𝟙_ C) X).inv := by
+  coherence
 #align category_theory.braiding_right_unitor_aux₁ CategoryTheory.braiding_rightUnitor_aux₁
 
 theorem braiding_rightUnitor_aux₂ (X : C) :
-    (𝟙 (𝟙_ C) ⊗ (β_ (𝟙_ C) X).hom) ≫ (𝟙 (𝟙_ C) ⊗ (ρ_ X).hom) = 𝟙 (𝟙_ C) ⊗ (λ_ X).hom :=
+    (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) = 𝟙_ C ◁ (λ_ X).hom :=
   calc
-    (𝟙 (𝟙_ C) ⊗ (β_ (𝟙_ C) X).hom) ≫ (𝟙 (𝟙_ C) ⊗ (ρ_ X).hom) =
-      (𝟙 (𝟙_ C) ⊗ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ (α_ _ _ _).hom ≫ (𝟙 (𝟙_ C) ⊗ (ρ_ X).hom) :=
+    (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (𝟙_ C ◁ (ρ_ X).hom) =
+      (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ (α_ _ _ _).hom ≫ (𝟙_ C ◁ (ρ_ X).hom) :=
       by coherence
-    _ = (𝟙 (𝟙_ C) ⊗ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ ((β_ _ X).hom ⊗ 𝟙 _) ≫
-          ((β_ _ X).inv ⊗ 𝟙 _) ≫ (α_ _ _ _).hom ≫ (𝟙 (𝟙_ C) ⊗ (ρ_ X).hom) :=
-      by (slice_rhs 3 4 => rw [← comp_tensor_id, Iso.hom_inv_id, tensor_id]); rw [id_comp]
-    _ = (α_ _ _ _).inv ≫ (β_ _ _).hom ≫ (α_ _ _ _).inv ≫ ((β_ _ X).inv ⊗ 𝟙 _) ≫ (α_ _ _ _).hom ≫
-          (𝟙 (𝟙_ C) ⊗ (ρ_ X).hom) :=
+    _ = (𝟙_ C ◁ (β_ (𝟙_ C) X).hom) ≫ (α_ _ _ _).inv ≫ ((β_ _ X).hom ▷ _) ≫
+          ((β_ _ X).inv ▷ _) ≫ (α_ _ _ _).hom ≫ (𝟙_ C ◁ (ρ_ X).hom) :=
+      by simp
+    _ = (α_ _ _ _).inv ≫ (β_ _ _).hom ≫ (α_ _ _ _).inv ≫ ((β_ _ X).inv ▷ _) ≫ (α_ _ _ _).hom ≫
+          (𝟙_ C ◁ (ρ_ X).hom) :=
       by (slice_lhs 1 3 => rw [← hexagon_reverse]); simp only [assoc]
-    _ = (α_ _ _ _).inv ≫ (β_ _ _).hom ≫ (𝟙 X ⊗ (ρ_ _).hom) ≫ (β_ _ X).inv :=
+    _ = (α_ _ _ _).inv ≫ (β_ _ _).hom ≫ (X ◁ (ρ_ _).hom) ≫ (β_ _ X).inv :=
       by rw [braiding_rightUnitor_aux₁]
-    _ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ⊗ 𝟙 _) ≫ (β_ _ X).hom ≫ (β_ _ _).inv :=
-      by (slice_lhs 2 3 => rw [← braiding_naturality]); simp only [assoc]
-    _ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ⊗ 𝟙 _) := by rw [Iso.hom_inv_id, comp_id]
-    _ = 𝟙 (𝟙_ C) ⊗ (λ_ X).hom := by rw [triangle_assoc_comp_right]
+    _ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ▷ _) ≫ (β_ _ X).hom ≫ (β_ _ _).inv :=
+      by (slice_lhs 2 3 => rw [← braiding_naturality_left]); simp only [assoc]
+    _ = (α_ _ _ _).inv ≫ ((ρ_ _).hom ▷ _) := by rw [Iso.hom_inv_id, comp_id]
+    _ = 𝟙_ C ◁ (λ_ X).hom := by rw [triangle_assoc_comp_right']
 
 #align category_theory.braiding_right_unitor_aux₂ CategoryTheory.braiding_rightUnitor_aux₂
 
-@[simp]
+@[reassoc]
 theorem braiding_rightUnitor (X : C) : (β_ (𝟙_ C) X).hom ≫ (ρ_ X).hom = (λ_ X).hom := by
-  rw [← tensor_left_iff, id_tensor_comp, braiding_rightUnitor_aux₂]
+  rw [← whiskerLeft_iff, MonoidalCategory.whiskerLeft_comp, braiding_rightUnitor_aux₂]
 #align category_theory.braiding_right_unitor CategoryTheory.braiding_rightUnitor
 
-@[simp]
+@[reassoc, simp]
+theorem braiding_tensorUnit_left (X : C) : (β_ (𝟙_ C) X).hom = (λ_ X).hom ≫ (ρ_ X).inv := by
+  simp [← braiding_rightUnitor]
+
+@[reassoc]
 theorem leftUnitor_inv_braiding (X : C) : (λ_ X).inv ≫ (β_ (𝟙_ C) X).hom = (ρ_ X).inv := by
-  apply (cancel_mono (ρ_ X).hom).1
-  simp only [assoc, braiding_rightUnitor, Iso.inv_hom_id]
+  simp
 #align category_theory.left_unitor_inv_braiding CategoryTheory.leftUnitor_inv_braiding
 
-@[simp]
+@[reassoc]
 theorem rightUnitor_inv_braiding (X : C) : (ρ_ X).inv ≫ (β_ X (𝟙_ C)).hom = (λ_ X).inv := by
   apply (cancel_mono (λ_ X).hom).1
   simp only [assoc, braiding_leftUnitor, Iso.inv_hom_id]
 #align category_theory.right_unitor_inv_braiding CategoryTheory.rightUnitor_inv_braiding
 
+@[reassoc, simp]
+theorem braiding_tensorUnit_right (X : C) : (β_ X (𝟙_ C)).hom = (ρ_ X).hom ≫ (λ_ X).inv := by
+  simp [← rightUnitor_inv_braiding]
+
 end
 
 /--
@@ -448,30 +454,20 @@ end CommMonoid
 section Tensor
 
 /-- The strength of the tensor product functor from `C × C` to `C`. -/
-def tensor_μ (X Y : C × C) : (tensor C).obj X ⊗ (tensor C).obj Y ⟶ (tensor C).obj (X ⊗ Y) :=
+def tensor_μ (X Y : C × C) : (X.1 ⊗ X.2) ⊗ Y.1 ⊗ Y.2 ⟶ (X.1 ⊗ Y.1) ⊗ X.2 ⊗ Y.2 :=
   (α_ X.1 X.2 (Y.1 ⊗ Y.2)).hom ≫
-    (𝟙 X.1 ⊗ (α_ X.2 Y.1 Y.2).inv) ≫
-      (𝟙 X.1 ⊗ (β_ X.2 Y.1).hom ⊗ 𝟙 Y.2) ≫
-        (𝟙 X.1 ⊗ (α_ Y.1 X.2 Y.2).hom) ≫ (α_ X.1 Y.1 (X.2 ⊗ Y.2)).inv
+    (X.1 ◁ (α_ X.2 Y.1 Y.2).inv) ≫
+      (X.1 ◁ (β_ X.2 Y.1).hom ▷ Y.2) ≫
+        (X.1 ◁ (α_ Y.1 X.2 Y.2).hom) ≫ (α_ X.1 Y.1 (X.2 ⊗ Y.2)).inv
 #align category_theory.tensor_μ CategoryTheory.tensor_μ
 
-theorem tensor_μ_def₁ (X₁ X₂ Y₁ Y₂ : C) :
-    tensor_μ C (X₁, X₂) (Y₁, Y₂) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) =
-      (α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ≫ (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ≫ (𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) :=
-  by dsimp [tensor_μ]; simp
-#align category_theory.tensor_μ_def₁ CategoryTheory.tensor_μ_def₁
-
-theorem tensor_μ_def₂ (X₁ X₂ Y₁ Y₂ : C) :
-    (𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).hom) ≫ (α_ X₁ X₂ (Y₁ ⊗ Y₂)).inv ≫ tensor_μ C (X₁, X₂) (Y₁, Y₂) =
-      (𝟙 X₁ ⊗ (β_ X₂ Y₁).hom ⊗ 𝟙 Y₂) ≫ (𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).hom) ≫ (α_ X₁ Y₁ (X₂ ⊗ Y₂)).inv :=
-  by dsimp [tensor_μ]; simp
-#align category_theory.tensor_μ_def₂ CategoryTheory.tensor_μ_def₂
-
+@[reassoc]
 theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : U₁ ⟶ V₁)
     (g₂ : U₂ ⟶ V₂) :
     ((f₁ ⊗ f₂) ⊗ g₁ ⊗ g₂) ≫ tensor_μ C (Y₁, Y₂) (V₁, V₂) =
       tensor_μ C (X₁, X₂) (U₁, U₂) ≫ ((f₁ ⊗ g₁) ⊗ f₂ ⊗ g₂) := by
-  dsimp [tensor_μ]
+  dsimp only [tensor_μ]
+  simp_rw [← id_tensorHom, ← tensorHom_id]
   slice_lhs 1 2 => rw [associator_naturality]
   slice_lhs 2 3 =>
     rw [← tensor_comp, comp_id f₁, ← id_comp f₁, associator_inv_naturality, tensor_comp]
@@ -483,301 +479,142 @@ theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ :
   simp only [assoc]
 #align category_theory.tensor_μ_natural CategoryTheory.tensor_μ_natural
 
+attribute [local simp] id_tensorHom tensorHom_id
+
 @[reassoc]
 theorem tensor_μ_natural_left {X₁ X₂ Y₁ Y₂ : C} (f₁: X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (Z₁ Z₂ : C) :
-    ((f₁ ⊗ f₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫ tensor_μ C (Y₁, Y₂) (Z₁, Z₂) =
-      tensor_μ C (X₁, X₂) (Z₁, Z₂) ≫ ((f₁ ⊗ 𝟙 Z₁) ⊗ (f₂ ⊗ 𝟙 Z₂)) := by
-  convert tensor_μ_natural C f₁ f₂ (𝟙 Z₁) (𝟙 Z₂) using 1; simp
+    (f₁ ⊗ f₂) ▷ (Z₁ ⊗ Z₂) ≫ tensor_μ C (Y₁, Y₂) (Z₁, Z₂) =
+      tensor_μ C (X₁, X₂) (Z₁, Z₂) ≫ (f₁ ▷ Z₁ ⊗ f₂ ▷ Z₂) := by
+  convert tensor_μ_natural C f₁ f₂ (𝟙 Z₁) (𝟙 Z₂) using 1 <;> simp
 
 @[reassoc]
 theorem tensor_μ_natural_right (Z₁ Z₂ : C) {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) :
-    (𝟙 (Z₁ ⊗ Z₂) ⊗ (f₁ ⊗ f₂)) ≫ tensor_μ C (Z₁, Z₂) (Y₁, Y₂) =
-      tensor_μ C (Z₁, Z₂) (X₁, X₂) ≫ ((𝟙 Z₁ ⊗ f₁) ⊗ (𝟙 Z₂ ⊗ f₂)) := by
-  convert tensor_μ_natural C (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1; simp
+    (Z₁ ⊗ Z₂) ◁ (f₁ ⊗ f₂) ≫ tensor_μ C (Z₁, Z₂) (Y₁, Y₂) =
+      tensor_μ C (Z₁, Z₂) (X₁, X₂) ≫ (Z₁ ◁ f₁ ⊗ Z₂ ◁ f₂) := by
+  convert tensor_μ_natural C (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1 <;> simp
 
+@[reassoc]
 theorem tensor_left_unitality (X₁ X₂ : C) :
     (λ_ (X₁ ⊗ X₂)).hom =
-      ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (X₁ ⊗ X₂)) ≫
+      ((λ_ (𝟙_ C)).inv ▷ (X₁ ⊗ X₂)) ≫
         tensor_μ C (𝟙_ C, 𝟙_ C) (X₁, X₂) ≫ ((λ_ X₁).hom ⊗ (λ_ X₂).hom) := by
-  dsimp [tensor_μ]
+  dsimp only [tensor_μ]
   have :
-    ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (X₁ ⊗ X₂)) ≫
-        (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).hom ≫ (𝟙 (𝟙_ C) ⊗ (α_ (𝟙_ C) X₁ X₂).inv) =
-      𝟙 (𝟙_ C) ⊗ (λ_ X₁).inv ⊗ 𝟙 X₂ :=
-    by pure_coherence
+    ((λ_ (𝟙_ C)).inv ▷ (X₁ ⊗ X₂)) ≫
+        (α_ (𝟙_ C) (𝟙_ C) (X₁ ⊗ X₂)).hom ≫ (𝟙_ C ◁ (α_ (𝟙_ C) X₁ X₂).inv) =
+      𝟙_ C ◁ (λ_ X₁).inv ▷ X₂ :=
+    by coherence
   slice_rhs 1 3 => rw [this]
   clear this
-  slice_rhs 1 2 => rw [← tensor_comp, ← tensor_comp, comp_id, comp_id, leftUnitor_inv_braiding]
-  simp only [assoc]
-  coherence
+  slice_rhs 1 2 => rw [← MonoidalCategory.whiskerLeft_comp, ← comp_whiskerRight,
+    leftUnitor_inv_braiding]
+  simp [tensorHom_id, id_tensorHom, tensorHom_def]
 #align category_theory.tensor_left_unitality CategoryTheory.tensor_left_unitality
 
+@[reassoc]
 theorem tensor_right_unitality (X₁ X₂ : C) :
     (ρ_ (X₁ ⊗ X₂)).hom =
-      (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).inv) ≫
+      ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).inv) ≫
         tensor_μ C (X₁, X₂) (𝟙_ C, 𝟙_ C) ≫ ((ρ_ X₁).hom ⊗ (ρ_ X₂).hom) := by
-  dsimp [tensor_μ]
+  dsimp only [tensor_μ]
   have :
-    (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).inv) ≫
-        (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).hom ≫ (𝟙 X₁ ⊗ (α_ X₂ (𝟙_ C) (𝟙_ C)).inv) =
-      (α_ X₁ X₂ (𝟙_ C)).hom ≫ (𝟙 X₁ ⊗ (ρ_ X₂).inv ⊗ 𝟙 (𝟙_ C)) :=
-    by pure_coherence
+    ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).inv) ≫
+        (α_ X₁ X₂ (𝟙_ C ⊗ 𝟙_ C)).hom ≫ (X₁ ◁ (α_ X₂ (𝟙_ C) (𝟙_ C)).inv) =
+      (α_ X₁ X₂ (𝟙_ C)).hom ≫ (X₁ ◁ (ρ_ X₂).inv ▷ 𝟙_ C) :=
+    by coherence
   slice_rhs 1 3 => rw [this]
   clear this
-  slice_rhs 2 3 => rw [← tensor_comp, ← tensor_comp, comp_id, comp_id, rightUnitor_inv_braiding]
-  simp only [assoc]
-  coherence
+  slice_rhs 2 3 => rw [← MonoidalCategory.whiskerLeft_comp, ← comp_whiskerRight,
+    rightUnitor_inv_braiding]
+  simp [tensorHom_id, id_tensorHom, tensorHom_def]
 #align category_theory.tensor_right_unitality CategoryTheory.tensor_right_unitality
 
-/-
-Diagram B6 from Proposition 1 of [Joyal and Street, *Braided monoidal categories*][Joyal_Street].
--/
-theorem tensor_associativity_aux (W X Y Z : C) :
-    ((β_ W X).hom ⊗ 𝟙 (Y ⊗ Z)) ≫
-        (α_ X W (Y ⊗ Z)).hom ≫
-          (𝟙 X ⊗ (α_ W Y Z).inv) ≫ (𝟙 X ⊗ (β_ (W ⊗ Y) Z).hom) ≫ (𝟙 X ⊗ (α_ Z W Y).inv) =
-      (𝟙 (W ⊗ X) ⊗ (β_ Y Z).hom) ≫
-        (α_ (W ⊗ X) Z Y).inv ≫
-          ((α_ W X Z).hom ⊗ 𝟙 Y) ≫
-            ((β_ W (X ⊗ Z)).hom ⊗ 𝟙 Y) ≫ ((α_ X Z W).hom ⊗ 𝟙 Y) ≫ (α_ X (Z ⊗ W) Y).hom := by
-  slice_rhs 3 5 => rw [← tensor_comp, ← tensor_comp, hexagon_forward, tensor_comp, tensor_comp]
-  slice_rhs 5 6 => rw [associator_naturality]
-  slice_rhs 2 3 => rw [← associator_inv_naturality]
-  slice_rhs 3 5 => rw [← pentagon_hom_inv]
-  slice_rhs 1 2 => rw [tensor_id, id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
-  slice_rhs 2 3 => rw [← tensor_id, associator_naturality]
-  slice_rhs 3 5 => rw [← tensor_comp, ← tensor_comp, ← hexagon_reverse, tensor_comp, tensor_comp]
-#align category_theory.tensor_associativity_aux CategoryTheory.tensor_associativity_aux
-
-set_option maxHeartbeats 400000 in
 theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
-    (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫
+    (tensor_μ C (X₁, X₂) (Y₁, Y₂) ▷ (Z₁ ⊗ Z₂)) ≫
         tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =
       (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫
-        (𝟙 (X₁ ⊗ X₂) ⊗ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) := by
-  have :
-    (α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom =
-      (α_ (X₁ ⊗ Y₁) Z₁ ((X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫
-        (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ Z₁ (X₂ ⊗ Y₂) Z₂).inv) ≫
-          (α_ X₁ Y₁ ((Z₁ ⊗ X₂ ⊗ Y₂) ⊗ Z₂)).hom ≫
-            (𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv) ≫
-              (α_ X₁ (Y₁ ⊗ Z₁ ⊗ X₂ ⊗ Y₂) Z₂).inv ≫
-                ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ X₂ Y₂).inv) ⊗ 𝟙 Z₂) ≫
-                  ((𝟙 X₁ ⊗ (α_ Y₁ (Z₁ ⊗ X₂) Y₂).inv) ⊗ 𝟙 Z₂) ≫
-                    ((𝟙 X₁ ⊗ (α_ Y₁ Z₁ X₂).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫
-                      (α_ X₁ (((Y₁ ⊗ Z₁) ⊗ X₂) ⊗ Y₂) Z₂).hom ≫
-                        (𝟙 X₁ ⊗ (α_ ((Y₁ ⊗ Z₁) ⊗ X₂) Y₂ Z₂).hom) ≫
-                          (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ Z₁) X₂ (Y₂ ⊗ Z₂)).hom) ≫
-                            (α_ X₁ (Y₁ ⊗ Z₁) (X₂ ⊗ Y₂ ⊗ Z₂)).inv :=
-    by pure_coherence
-  rw [this]; clear this
-  slice_lhs 2 4 => rw [tensor_μ_def₁]
-  slice_lhs 4 5 => rw [← tensor_id, associator_naturality]
-  slice_lhs 5 6 => rw [← tensor_comp, associator_inv_naturality, tensor_comp]
-  slice_lhs 6 7 => rw [associator_inv_naturality]
-  have :
-    (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (Z₁ ⊗ Z₂)).hom ≫
-        (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ (X₂ ⊗ Y₂) Z₁ Z₂).inv) ≫
-          (α_ X₁ Y₁ (((X₂ ⊗ Y₂) ⊗ Z₁) ⊗ Z₂)).hom ≫
-            (𝟙 X₁ ⊗ (α_ Y₁ ((X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv) ≫ (α_ X₁ (Y₁ ⊗ (X₂ ⊗ Y₂) ⊗ Z₁) Z₂).inv =
-      ((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫
-        ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫
-          (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) Z₁ Z₂).inv ≫
-            ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) Z₁).hom ⊗ 𝟙 Z₂) ≫
-              ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ Z₁).hom) ⊗ 𝟙 Z₂) ≫
-                ((𝟙 X₁ ⊗ (α_ Y₁ X₂ (Y₂ ⊗ Z₁)).hom) ⊗ 𝟙 Z₂) ≫
-                  ((𝟙 X₁ ⊗ 𝟙 Y₁ ⊗ (α_ X₂ Y₂ Z₁).inv) ⊗ 𝟙 Z₂) :=
-    by pure_coherence
-  slice_lhs 2 6 => rw [this]
-  clear this
-  slice_lhs 1 3 => rw [← tensor_comp, ← tensor_comp, tensor_μ_def₁, tensor_comp, tensor_comp]
-  slice_lhs 3 4 => rw [← tensor_id, associator_inv_naturality]
-  slice_lhs 4 5 => rw [← tensor_comp, associator_naturality, tensor_comp]
-  slice_lhs 5 6 =>
-    rw [← tensor_comp, ← tensor_comp, associator_naturality, tensor_comp, tensor_comp]
-  slice_lhs 6 10 =>
-    rw [← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp, ←
-      tensor_comp, ← tensor_comp, tensor_id, tensor_associativity_aux, ← tensor_id, ←
-      id_comp (𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁), ← id_comp (𝟙 Z₂ ≫ 𝟙 Z₂ ≫ 𝟙 Z₂ ≫ 𝟙 Z₂ ≫ 𝟙 Z₂),
-      tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp,
-      tensor_comp, tensor_comp, tensor_comp]
-  slice_lhs 11 12 =>
-    rw [← tensor_comp, ← tensor_comp, Iso.hom_inv_id]
-    simp
-  simp only [assoc, id_comp]
-  slice_lhs 10 11 =>
-    rw [← tensor_comp, ← tensor_comp, ← tensor_comp, Iso.hom_inv_id]
-    simp
-  simp only [assoc, id_comp]
-  slice_lhs 9 10 => rw [associator_naturality]
-  slice_lhs 10 11 => rw [← tensor_comp, associator_naturality, tensor_comp]
-  slice_lhs 11 13 => rw [tensor_id, ← tensor_μ_def₂]
-  have :
-    ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Z₁ Y₂).inv) ⊗ 𝟙 Z₂) ≫
-        ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Z₁).hom ⊗ 𝟙 Y₂) ⊗ 𝟙 Z₂) ≫
-          (α_ X₁ ((X₂ ⊗ Y₁ ⊗ Z₁) ⊗ Y₂) Z₂).hom ≫
-            (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁ ⊗ Z₁) Y₂ Z₂).hom) ≫
-              (𝟙 X₁ ⊗ (α_ X₂ (Y₁ ⊗ Z₁) (Y₂ ⊗ Z₂)).hom) ≫ (α_ X₁ X₂ ((Y₁ ⊗ Z₁) ⊗ Y₂ ⊗ Z₂)).inv =
-      (α_ X₁ ((X₂ ⊗ Y₁) ⊗ Z₁ ⊗ Y₂) Z₂).hom ≫
-        (𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) (Z₁ ⊗ Y₂) Z₂).hom) ≫
-          (𝟙 X₁ ⊗ (α_ X₂ Y₁ ((Z₁ ⊗ Y₂) ⊗ Z₂)).hom) ≫
-            (α_ X₁ X₂ (Y₁ ⊗ (Z₁ ⊗ Y₂) ⊗ Z₂)).inv ≫
-              (𝟙 (X₁ ⊗ X₂) ⊗ 𝟙 Y₁ ⊗ (α_ Z₁ Y₂ Z₂).hom) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (α_ Y₁ Z₁ (Y₂ ⊗ Z₂)).inv) :=
-    by pure_coherence
-  slice_lhs 7 12 => rw [this]
-  clear this
-  slice_lhs 6 7 => rw [associator_naturality]
-  slice_lhs 7 8 => rw [← tensor_comp, associator_naturality, tensor_comp]
-  slice_lhs 8 9 => rw [← tensor_comp, associator_naturality, tensor_comp]
-  slice_lhs 9 10 => rw [associator_inv_naturality]
-  slice_lhs 10 12 => rw [← tensor_comp, ← tensor_comp, ← tensor_μ_def₂, tensor_comp, tensor_comp]
-  dsimp
-  coherence
+        ((X₁ ⊗ X₂) ◁ tensor_μ C (Y₁, Y₂) (Z₁, Z₂)) ≫ tensor_μ C (X₁, X₂) (Y₁ ⊗ Z₁, Y₂ ⊗ Z₂) := by
+  dsimp only [tensor_obj, prodMonoidal_tensorObj, tensor_μ]
+  simp only [whiskerRight_tensor, comp_whiskerRight, whisker_assoc, assoc, Iso.inv_hom_id_assoc,
+    tensor_whiskerLeft, braiding_tensor_left, MonoidalCategory.whiskerLeft_comp,
+    braiding_tensor_right]
+  calc
+    _ = 𝟙 _ ⊗≫
+      X₁ ◁ ((β_ X₂ Y₁).hom ▷ (Y₂ ⊗ Z₁) ≫ (Y₁ ⊗ X₂) ◁ (β_ Y₂ Z₁).hom) ▷ Z₂ ⊗≫
+        X₁ ◁ Y₁ ◁ (β_ X₂ Z₁).hom ▷ Y₂ ▷ Z₂ ⊗≫ 𝟙 _ := by coherence
+    _ = _ := by rw [← whisker_exchange]; coherence
 #align category_theory.tensor_associativity CategoryTheory.tensor_associativity
 
+-- We got a timeout if `reassoc` was at the declaration, so we put it here instead.
+attribute [reassoc] tensor_associativity
+
 /-- The tensor product functor from `C × C` to `C` as a monoidal functor. -/
 @[simps!]
 def tensorMonoidal : MonoidalFunctor (C × C) C :=
   { tensor C with
     ε := (λ_ (𝟙_ C)).inv
     μ := tensor_μ C
-    μ_natural_left := fun f Z => tensor_μ_natural_left C f.1 f.2 Z.1 Z.2
-    μ_natural_right := fun Z f => tensor_μ_natural_right C Z.1 Z.2 f.1 f.2
-    associativity := fun X Y Z => tensor_associativity C X.1 X.2 Y.1 Y.2 Z.1 Z.2
-    left_unitality := fun ⟨X₁, X₂⟩ => tensor_left_unitality C X₁ X₂
-    right_unitality := fun ⟨X₁, X₂⟩ => tensor_right_unitality C X₁ X₂
+    μ_natural_left := fun f Z => by
+      -- `simpa` will be not needed when we define `μ_natural_left` in terms of the whiskerings.
+      simpa using tensor_μ_natural_left C f.1 f.2 Z.1 Z.2
+    μ_natural_right := fun Z f => by
+      simpa using tensor_μ_natural_right C Z.1 Z.2 f.1 f.2
+    associativity := fun X Y Z => by
+      simpa using tensor_associativity C X.1 X.2 Y.1 Y.2 Z.1 Z.2
+    left_unitality := fun ⟨X₁, X₂⟩ => by
+      simpa using tensor_left_unitality C X₁ X₂
+    right_unitality := fun ⟨X₁, X₂⟩ => by
+      simpa using tensor_right_unitality C X₁ X₂
     μ_isIso := by dsimp [tensor_μ]; infer_instance }
-#align category_theory.tensor_monoidal CategoryTheory.tensorMonoidal
 
+@[reassoc]
 theorem leftUnitor_monoidal (X₁ X₂ : C) :
     (λ_ X₁).hom ⊗ (λ_ X₂).hom =
-      tensor_μ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ ((λ_ (𝟙_ C)).hom ⊗ 𝟙 (X₁ ⊗ X₂)) ≫ (λ_ (X₁ ⊗ X₂)).hom := by
-  dsimp [tensor_μ]
+      tensor_μ C (𝟙_ C, X₁) (𝟙_ C, X₂) ≫ ((λ_ (𝟙_ C)).hom ▷ (X₁ ⊗ X₂)) ≫ (λ_ (X₁ ⊗ X₂)).hom := by
+  dsimp only [tensor_μ]
   have :
     (λ_ X₁).hom ⊗ (λ_ X₂).hom =
       (α_ (𝟙_ C) X₁ (𝟙_ C ⊗ X₂)).hom ≫
-        (𝟙 (𝟙_ C) ⊗ (α_ X₁ (𝟙_ C) X₂).inv) ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ ((ρ_ X₁).hom ⊗ 𝟙 X₂) :=
-    by pure_coherence
+        (𝟙_ C ◁ (α_ X₁ (𝟙_ C) X₂).inv) ≫ (λ_ ((X₁ ⊗ 𝟙_ C) ⊗ X₂)).hom ≫ ((ρ_ X₁).hom ▷ X₂) :=
+    by coherence
   rw [this]; clear this
   rw [← braiding_leftUnitor]
-  slice_lhs 3 4 => rw [← id_comp (𝟙 X₂), tensor_comp]
-  slice_lhs 3 4 => rw [← leftUnitor_naturality]
+  dsimp only [tensor_obj, prodMonoidal_tensorObj]
   coherence
 #align category_theory.left_unitor_monoidal CategoryTheory.leftUnitor_monoidal
 
+@[reassoc]
 theorem rightUnitor_monoidal (X₁ X₂ : C) :
     (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
-      tensor_μ C (X₁, 𝟙_ C) (X₂, 𝟙_ C) ≫ (𝟙 (X₁ ⊗ X₂) ⊗ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (X₁ ⊗ X₂)).hom := by
-  dsimp [tensor_μ]
+      tensor_μ C (X₁, 𝟙_ C) (X₂, 𝟙_ C) ≫ ((X₁ ⊗ X₂) ◁ (λ_ (𝟙_ C)).hom) ≫ (ρ_ (X₁ ⊗ X₂)).hom := by
+  dsimp only [tensor_μ]
   have :
     (ρ_ X₁).hom ⊗ (ρ_ X₂).hom =
       (α_ X₁ (𝟙_ C) (X₂ ⊗ 𝟙_ C)).hom ≫
-        (𝟙 X₁ ⊗ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv) ≫ (𝟙 X₁ ⊗ (ρ_ (𝟙_ C ⊗ X₂)).hom) ≫ (𝟙 X₁ ⊗ (λ_ X₂).hom) :=
-    by pure_coherence
+        (X₁ ◁ (α_ (𝟙_ C) X₂ (𝟙_ C)).inv) ≫ (X₁ ◁ (ρ_ (𝟙_ C ⊗ X₂)).hom) ≫ (X₁ ◁ (λ_ X₂).hom) :=
+    by coherence
   rw [this]; clear this
   rw [← braiding_rightUnitor]
-  slice_lhs 3 4 => rw [← id_comp (𝟙 X₁), tensor_comp, id_comp]
-  slice_lhs 3 4 => rw [← tensor_comp, ← rightUnitor_naturality, tensor_comp]
+  dsimp only [tensor_obj, prodMonoidal_tensorObj]
   coherence
 #align category_theory.right_unitor_monoidal CategoryTheory.rightUnitor_monoidal
 
-theorem associator_monoidal_aux (W X Y Z : C) :
-    (𝟙 W ⊗ (β_ X (Y ⊗ Z)).hom) ≫
-        (𝟙 W ⊗ (α_ Y Z X).hom) ≫ (α_ W Y (Z ⊗ X)).inv ≫ ((β_ W Y).hom ⊗ 𝟙 (Z ⊗ X)) =
-      (α_ W X (Y ⊗ Z)).inv ≫
-        (α_ (W ⊗ X) Y Z).inv ≫
-          ((β_ (W ⊗ X) Y).hom ⊗ 𝟙 Z) ≫
-            ((α_ Y W X).inv ⊗ 𝟙 Z) ≫ (α_ (Y ⊗ W) X Z).hom ≫ (𝟙 (Y ⊗ W) ⊗ (β_ X Z).hom) := by
-  slice_rhs 1 2 => rw [← pentagon_inv]
-  slice_rhs 3 5 => rw [← tensor_comp, ← tensor_comp, hexagon_reverse, tensor_comp, tensor_comp]
-  slice_rhs 5 6 => rw [associator_naturality]
-  slice_rhs 6 7 => rw [tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
-  slice_rhs 2 3 => rw [← associator_inv_naturality]
-  slice_rhs 3 5 => rw [pentagon_inv_inv_hom]
-  slice_rhs 4 5 => rw [← tensor_id, ← associator_inv_naturality]
-  slice_rhs 2 4 => rw [← tensor_comp, ← tensor_comp, ← hexagon_forward, tensor_comp, tensor_comp]
-  simp
-#align category_theory.associator_monoidal_aux CategoryTheory.associator_monoidal_aux
-
-set_option maxHeartbeats 400000 in
 theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ : C) :
     tensor_μ C (X₁ ⊗ X₂, X₃) (Y₁ ⊗ Y₂, Y₃) ≫
-        (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =
+        (tensor_μ C (X₁, X₂) (Y₁, Y₂) ▷ (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =
       ((α_ X₁ X₂ X₃).hom ⊗ (α_ Y₁ Y₂ Y₃).hom) ≫
-        tensor_μ C (X₁, X₂ ⊗ X₃) (Y₁, Y₂ ⊗ Y₃) ≫ (𝟙 (X₁ ⊗ Y₁) ⊗ tensor_μ C (X₂, X₃) (Y₂, Y₃)) := by
-  have :
-    (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =
-      ((α_ X₁ Y₁ (X₂ ⊗ Y₂)).hom ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫
-        ((𝟙 X₁ ⊗ (α_ Y₁ X₂ Y₂).inv) ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫
-          (α_ (X₁ ⊗ (Y₁ ⊗ X₂) ⊗ Y₂) X₃ Y₃).inv ≫
-            ((α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂) X₃).hom ⊗ 𝟙 Y₃) ≫
-              ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) Y₂ X₃).hom) ⊗ 𝟙 Y₃) ≫
-                (α_ X₁ ((Y₁ ⊗ X₂) ⊗ Y₂ ⊗ X₃) Y₃).hom ≫
-                  (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) (Y₂ ⊗ X₃) Y₃).hom) ≫
-                    (𝟙 X₁ ⊗ (α_ Y₁ X₂ ((Y₂ ⊗ X₃) ⊗ Y₃)).hom) ≫
-                      (α_ X₁ Y₁ (X₂ ⊗ (Y₂ ⊗ X₃) ⊗ Y₃)).inv ≫
-                        (𝟙 (X₁ ⊗ Y₁) ⊗ 𝟙 X₂ ⊗ (α_ Y₂ X₃ Y₃).hom) ≫
-                          (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ X₂ Y₂ (X₃ ⊗ Y₃)).inv) :=
-    by pure_coherence
-  rw [this]; clear this
-  slice_lhs 2 4 => rw [← tensor_comp, ← tensor_comp, tensor_μ_def₁, tensor_comp, tensor_comp]
-  slice_lhs 4 5 => rw [← tensor_id, associator_inv_naturality]
-  slice_lhs 5 6 => rw [← tensor_comp, associator_naturality, tensor_comp]
-  slice_lhs 6 7 =>
-    rw [← tensor_comp, ← tensor_comp, associator_naturality, tensor_comp, tensor_comp]
-  have :
-    ((α_ X₁ X₂ (Y₁ ⊗ Y₂)).hom ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫
-        ((𝟙 X₁ ⊗ (α_ X₂ Y₁ Y₂).inv) ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫
-          (α_ (X₁ ⊗ (X₂ ⊗ Y₁) ⊗ Y₂) X₃ Y₃).inv ≫
-            ((α_ X₁ ((X₂ ⊗ Y₁) ⊗ Y₂) X₃).hom ⊗ 𝟙 Y₃) ≫ ((𝟙 X₁ ⊗ (α_ (X₂ ⊗ Y₁) Y₂ X₃).hom) ⊗ 𝟙 Y₃) =
-      (α_ (X₁ ⊗ X₂) (Y₁ ⊗ Y₂) (X₃ ⊗ Y₃)).hom ≫
-        (𝟙 (X₁ ⊗ X₂) ⊗ (α_ (Y₁ ⊗ Y₂) X₃ Y₃).inv) ≫
-          (α_ X₁ X₂ (((Y₁ ⊗ Y₂) ⊗ X₃) ⊗ Y₃)).hom ≫
-            (𝟙 X₁ ⊗ (α_ X₂ ((Y₁ ⊗ Y₂) ⊗ X₃) Y₃).inv) ≫
-              (α_ X₁ (X₂ ⊗ (Y₁ ⊗ Y₂) ⊗ X₃) Y₃).inv ≫
-                ((𝟙 X₁ ⊗ 𝟙 X₂ ⊗ (α_ Y₁ Y₂ X₃).hom) ⊗ 𝟙 Y₃) ≫
-                  ((𝟙 X₁ ⊗ (α_ X₂ Y₁ (Y₂ ⊗ X₃)).inv) ⊗ 𝟙 Y₃) :=
-    by pure_coherence
-  slice_lhs 2 6 => rw [this]
-  clear this
-  slice_lhs 1 3 => rw [tensor_μ_def₁]
-  slice_lhs 3 4 => rw [← tensor_id, associator_naturality]
-  slice_lhs 4 5 => rw [← tensor_comp, associator_inv_naturality, tensor_comp]
-  slice_lhs 5 6 => rw [associator_inv_naturality]
-  slice_lhs 6 9 =>
-    rw [← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp, ← tensor_comp,
-      tensor_id, associator_monoidal_aux, ← id_comp (𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁), ←
-      id_comp (𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁ ≫ 𝟙 X₁), ← id_comp (𝟙 Y₃ ≫ 𝟙 Y₃ ≫ 𝟙 Y₃ ≫ 𝟙 Y₃), ←
-      id_comp (𝟙 Y₃ ≫ 𝟙 Y₃ ≫ 𝟙 Y₃ ≫ 𝟙 Y₃ ≫ 𝟙 Y₃), tensor_comp, tensor_comp, tensor_comp,
-      tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp, tensor_comp]
-  slice_lhs 11 12 => rw [associator_naturality]
-  slice_lhs 12 13 => rw [← tensor_comp, associator_naturality, tensor_comp]
-  slice_lhs 13 14 => rw [← tensor_comp, ← tensor_id, associator_naturality, tensor_comp]
-  slice_lhs 14 15 => rw [associator_inv_naturality]
-  slice_lhs 15 17 =>
-    rw [tensor_id, ← tensor_comp, ← tensor_comp, ← tensor_μ_def₂, tensor_comp, tensor_comp]
-  have :
-    ((𝟙 X₁ ⊗ (α_ Y₁ X₂ X₃).inv ⊗ 𝟙 Y₂) ⊗ 𝟙 Y₃) ≫
-        ((𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) X₃ Y₂).hom) ⊗ 𝟙 Y₃) ≫
-          (α_ X₁ ((Y₁ ⊗ X₂) ⊗ X₃ ⊗ Y₂) Y₃).hom ≫
-            (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂) (X₃ ⊗ Y₂) Y₃).hom) ≫
-              (𝟙 X₁ ⊗ (α_ Y₁ X₂ ((X₃ ⊗ Y₂) ⊗ Y₃)).hom) ≫
-                (α_ X₁ Y₁ (X₂ ⊗ (X₃ ⊗ Y₂) ⊗ Y₃)).inv ≫
-                  (𝟙 (X₁ ⊗ Y₁) ⊗ 𝟙 X₂ ⊗ (α_ X₃ Y₂ Y₃).hom) ≫
-                    (𝟙 (X₁ ⊗ Y₁) ⊗ (α_ X₂ X₃ (Y₂ ⊗ Y₃)).inv) =
-      (α_ X₁ ((Y₁ ⊗ X₂ ⊗ X₃) ⊗ Y₂) Y₃).hom ≫
-        (𝟙 X₁ ⊗ (α_ (Y₁ ⊗ X₂ ⊗ X₃) Y₂ Y₃).hom) ≫
-          (𝟙 X₁ ⊗ (α_ Y₁ (X₂ ⊗ X₃) (Y₂ ⊗ Y₃)).hom) ≫ (α_ X₁ Y₁ ((X₂ ⊗ X₃) ⊗ Y₂ ⊗ Y₃)).inv :=
-    by pure_coherence
-  slice_lhs 9 16 => rw [this]
-  clear this
-  slice_lhs 8 9 => rw [associator_naturality]
-  slice_lhs 9 10 => rw [← tensor_comp, associator_naturality, tensor_comp]
-  slice_lhs 10 12 => rw [tensor_id, ← tensor_μ_def₂]
-  dsimp
-  coherence
+        tensor_μ C (X₁, X₂ ⊗ X₃) (Y₁, Y₂ ⊗ Y₃) ≫ ((X₁ ⊗ Y₁) ◁ tensor_μ C (X₂, X₃) (Y₂, Y₃)) := by
+  dsimp only [tensor_μ]
+  calc
+    _ = 𝟙 _ ⊗≫ X₁ ◁ X₂ ◁ (β_ X₃ Y₁).hom ▷ Y₂ ▷ Y₃ ⊗≫
+      X₁ ◁ ((X₂ ⊗ Y₁) ◁ (β_ X₃ Y₂).hom ≫
+        (β_ X₂ Y₁).hom ▷ (Y₂ ⊗ X₃)) ▷ Y₃ ⊗≫ 𝟙 _ := by simp; coherence
+    _ = _ := by rw [whisker_exchange]; simp; coherence
 #align category_theory.associator_monoidal CategoryTheory.associator_monoidal
 
+-- We got a timeout if `reassoc` was at the declaration, so we put it here instead.
+attribute [reassoc] associator_monoidal
+
 end Tensor
 
 end CategoryTheory

Extracted from #6307. The main reason why #6307 is so large is that many tensoring of identity morphisms that appear in mathlib should be replaced with whiskerings. This PR will leave this issue and deal with other parts. That is, we do not set id_tensorHom and tensorHom_id as simple lemmas at this moment, We can set them as simp lemmas locally to enable simple normal forms.

@@ -202,7 +202,7 @@ theorem braiding_leftUnitor_aux₁ (X : C) :
     (α_ (𝟙_ C) (𝟙_ C) X).hom ≫
         (𝟙 (𝟙_ C) ⊗ (β_ X (𝟙_ C)).inv) ≫ (α_ _ X _).inv ≫ ((λ_ X).hom ⊗ 𝟙 _) =
       ((λ_ _).hom ⊗ 𝟙 X) ≫ (β_ X (𝟙_ C)).inv :=
-  by rw [← leftUnitor_tensor, leftUnitor_naturality]; simp
+  by rw [← leftUnitor_tensor, leftUnitor_naturality]; simp [id_tensorHom, tensorHom_id]
 #align category_theory.braiding_left_unitor_aux₁ CategoryTheory.braiding_leftUnitor_aux₁
 
 theorem braiding_leftUnitor_aux₂ (X : C) :
@@ -235,7 +235,7 @@ theorem braiding_rightUnitor_aux₁ (X : C) :
     (α_ X (𝟙_ C) (𝟙_ C)).inv ≫
         ((β_ (𝟙_ C) X).inv ⊗ 𝟙 (𝟙_ C)) ≫ (α_ _ X _).hom ≫ (𝟙 _ ⊗ (ρ_ X).hom) =
       (𝟙 X ⊗ (ρ_ _).hom) ≫ (β_ (𝟙_ C) X).inv :=
-  by rw [← rightUnitor_tensor, rightUnitor_naturality]; simp
+  by rw [← rightUnitor_tensor, rightUnitor_naturality]; simp [id_tensorHom, tensorHom_id]
 #align category_theory.braiding_right_unitor_aux₁ CategoryTheory.braiding_rightUnitor_aux₁
 
 theorem braiding_rightUnitor_aux₂ (X : C) :

Make all the notations that unambiguously should inherit the docstring of their definition actually inherit it.

Also write a few docstrings by hand. I only wrote the ones I was competent to write and which I was sure of. Some docstrings come from mathlib3 as they were lost during the early port.

This PR is only intended as a first pass There are many more docstrings to add.

@@ -40,7 +40,7 @@ which is natural in both arguments,
 and also satisfies the two hexagon identities.
 -/
 class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] where
-  /-- braiding natural isomorphism -/
+  /-- The braiding natural isomorphism. -/
   braiding : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X
   -- Note: `𝟙 X ⊗ f` will be replaced by `X ◁ f` (and similarly for `f ⊗ 𝟙 Z`) in #6307.
   braiding_naturality_right :
@@ -51,12 +51,13 @@ class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] whe
     ∀ {X Y : C} (f : X ⟶ Y) (Z : C),
       (f ⊗ 𝟙 Z) ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ (𝟙 Z ⊗ f) := by
     aesop_cat
-  -- hexagon identities:
+  /-- The first hexagon identity. -/
   hexagon_forward :
     ∀ X Y Z : C,
       (α_ X Y Z).hom ≫ (braiding X (Y ⊗ Z)).hom ≫ (α_ Y Z X).hom =
         ((braiding X Y).hom ⊗ 𝟙 Z) ≫ (α_ Y X Z).hom ≫ (𝟙 Y ⊗ (braiding X Z).hom) := by
     aesop_cat
+  /-- The second hexagon identity. -/
   hexagon_reverse :
     ∀ X Y Z : C,
       (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =
@@ -75,6 +76,7 @@ open MonoidalCategory
 
 open BraidedCategory
 
+@[inherit_doc]
 notation "β_" => BraidedCategory.braiding
 
 namespace BraidedCategory

Extracted from #6307. We replace some axioms by those more preferable when using the whiskerings instead of the tensor of morphisms.

@@ -40,11 +40,16 @@ which is natural in both arguments,
 and also satisfies the two hexagon identities.
 -/
 class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] where
-  -- braiding natural iso:
+  /-- braiding natural isomorphism -/
   braiding : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X
-  braiding_naturality :
-    ∀ {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y'),
-      (f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) := by
+  -- Note: `𝟙 X ⊗ f` will be replaced by `X ◁ f` (and similarly for `f ⊗ 𝟙 Z`) in #6307.
+  braiding_naturality_right :
+    ∀ (X : C) {Y Z : C} (f : Y ⟶ Z),
+      (𝟙 X ⊗ f) ≫ (braiding X Z).hom = (braiding X Y).hom ≫ (f ⊗ 𝟙 X) := by
+    aesop_cat
+  braiding_naturality_left :
+    ∀ {X Y : C} (f : X ⟶ Y) (Z : C),
+      (f ⊗ 𝟙 Z) ≫ (braiding Y Z).hom = (braiding X Z).hom ≫ (𝟙 Z ⊗ f) := by
     aesop_cat
   -- hexagon identities:
   hexagon_forward :
@@ -59,7 +64,9 @@ class BraidedCategory (C : Type u) [Category.{v} C] [MonoidalCategory.{v} C] whe
     aesop_cat
 #align category_theory.braided_category CategoryTheory.BraidedCategory
 
-attribute [reassoc (attr := simp)] BraidedCategory.braiding_naturality
+attribute [reassoc (attr := simp)]
+  BraidedCategory.braiding_naturality_left
+  BraidedCategory.braiding_naturality_right
 attribute [reassoc] BraidedCategory.hexagon_forward BraidedCategory.hexagon_reverse
 
 open Category
@@ -68,7 +75,54 @@ open MonoidalCategory
 
 open BraidedCategory
 
-notation "β_" => braiding
+notation "β_" => BraidedCategory.braiding
+
+namespace BraidedCategory
+
+variable {C : Type u} [Category.{v} C] [MonoidalCategory.{v} C] [BraidedCategory.{v} C]
+
+@[simp]
+theorem braiding_tensor_left (X Y Z : C) :
+    (β_ (X ⊗ Y) Z).hom  =
+      (α_ X Y Z).hom ≫ (𝟙 X ⊗ (β_ Y Z).hom) ≫ (α_ X Z Y).inv ≫
+        ((β_ X Z).hom ⊗ 𝟙 Y) ≫ (α_ Z X Y).hom := by
+  intros
+  apply (cancel_epi (α_ X Y Z).inv).1
+  apply (cancel_mono (α_ Z X Y).inv).1
+  simp [hexagon_reverse]
+
+@[simp]
+theorem braiding_tensor_right (X Y Z : C) :
+    (β_ X (Y ⊗ Z)).hom  =
+      (α_ X Y Z).inv ≫ ((β_ X Y).hom ⊗ 𝟙 Z) ≫ (α_ Y X Z).hom ≫
+        (𝟙 Y ⊗ (β_ X Z).hom) ≫ (α_ Y Z X).inv := by
+  intros
+  apply (cancel_epi (α_ X Y Z).hom).1
+  apply (cancel_mono (α_ Y Z X).hom).1
+  simp [hexagon_forward]
+
+@[simp]
+theorem braiding_inv_tensor_left (X Y Z : C) :
+    (β_ (X ⊗ Y) Z).inv  =
+      (α_ Z X Y).inv ≫ ((β_ X Z).inv ⊗ 𝟙 Y) ≫ (α_ X Z Y).hom ≫
+        (𝟙 X ⊗ (β_ Y Z).inv) ≫ (α_ X Y Z).inv :=
+  eq_of_inv_eq_inv (by simp)
+
+@[simp]
+theorem braiding_inv_tensor_right (X Y Z : C) :
+    (β_ X (Y ⊗ Z)).inv  =
+      (α_ Y Z X).hom ≫ (𝟙 Y ⊗ (β_ X Z).inv) ≫ (α_ Y X Z).inv ≫
+        ((β_ X Y).inv ⊗ 𝟙 Z) ≫ (α_ X Y Z).hom :=
+  eq_of_inv_eq_inv (by simp)
+
+-- The priority setting will not be needed when we replace `𝟙 X ⊗ f` by `X ◁ f`.
+@[reassoc (attr := simp (low))]
+theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
+    (f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) := by
+  rw [← tensor_id_comp_id_tensor f g, ← id_tensor_comp_tensor_id g f]
+  simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc]
+
+end BraidedCategory
 
 /--
 Verifying the axioms for a braiding by checking that the candidate braiding is sent to a braiding
@@ -79,12 +133,18 @@ def braidedCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalC
     (β : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X)
     (w : ∀ X Y, F.μ _ _ ≫ F.map (β X Y).hom = (β_ _ _).hom ≫ F.μ _ _) : BraidedCategory C where
   braiding := β
-  braiding_naturality := by
+  braiding_naturality_left := by
+    intros
+    apply F.map_injective
+    refine (cancel_epi (F.μ ?_ ?_)).1 ?_
+    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_left_assoc, w, Functor.map_comp,
+      reassoc_of% w, braiding_naturality_left_assoc, LaxMonoidalFunctor.μ_natural_right]
+  braiding_naturality_right := by
     intros
     apply F.map_injective
     refine (cancel_epi (F.μ ?_ ?_)).1 ?_
-    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_assoc, w, Functor.map_comp, reassoc_of% w,
-      braiding_naturality_assoc, LaxMonoidalFunctor.μ_natural]
+    rw [Functor.map_comp, ← LaxMonoidalFunctor.μ_natural_right_assoc, w, Functor.map_comp,
+      reassoc_of% w, braiding_naturality_right_assoc, LaxMonoidalFunctor.μ_natural_left]
   hexagon_forward := by
     intros
     apply F.map_injective

Extracted from #6307. We replace μ_natural with μ_natural_left and μ_natural_right since we prefer to use the whiskerings to the tensor of morphisms in the refactor #6307.

@@ -421,6 +421,18 @@ theorem tensor_μ_natural {X₁ X₂ Y₁ Y₂ U₁ U₂ V₁ V₂ : C} (f₁ :
   simp only [assoc]
 #align category_theory.tensor_μ_natural CategoryTheory.tensor_μ_natural
 
+@[reassoc]
+theorem tensor_μ_natural_left {X₁ X₂ Y₁ Y₂ : C} (f₁: X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (Z₁ Z₂ : C) :
+    ((f₁ ⊗ f₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫ tensor_μ C (Y₁, Y₂) (Z₁, Z₂) =
+      tensor_μ C (X₁, X₂) (Z₁, Z₂) ≫ ((f₁ ⊗ 𝟙 Z₁) ⊗ (f₂ ⊗ 𝟙 Z₂)) := by
+  convert tensor_μ_natural C f₁ f₂ (𝟙 Z₁) (𝟙 Z₂) using 1; simp
+
+@[reassoc]
+theorem tensor_μ_natural_right (Z₁ Z₂ : C) {X₁ X₂ Y₁ Y₂ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) :
+    (𝟙 (Z₁ ⊗ Z₂) ⊗ (f₁ ⊗ f₂)) ≫ tensor_μ C (Z₁, Z₂) (Y₁, Y₂) =
+      tensor_μ C (Z₁, Z₂) (X₁, X₂) ≫ ((𝟙 Z₁ ⊗ f₁) ⊗ (𝟙 Z₂ ⊗ f₂)) := by
+  convert tensor_μ_natural C (𝟙 Z₁) (𝟙 Z₂) f₁ f₂ using 1; simp
+
 theorem tensor_left_unitality (X₁ X₂ : C) :
     (λ_ (X₁ ⊗ X₂)).hom =
       ((λ_ (𝟙_ C)).inv ⊗ 𝟙 (X₁ ⊗ X₂)) ≫
@@ -566,8 +578,9 @@ theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
 def tensorMonoidal : MonoidalFunctor (C × C) C :=
   { tensor C with
     ε := (λ_ (𝟙_ C)).inv
-    μ := fun X Y => tensor_μ C X Y
-    μ_natural := fun f g => tensor_μ_natural C f.1 f.2 g.1 g.2
+    μ := tensor_μ C
+    μ_natural_left := fun f Z => tensor_μ_natural_left C f.1 f.2 Z.1 Z.2
+    μ_natural_right := fun Z f => tensor_μ_natural_right C Z.1 Z.2 f.1 f.2
     associativity := fun X Y Z => tensor_associativity C X.1 X.2 Y.1 Y.2 Z.1 Z.2
     left_unitality := fun ⟨X₁, X₂⟩ => tensor_left_unitality C X₁ X₂
     right_unitality := fun ⟨X₁, X₂⟩ => tensor_right_unitality C X₁ X₂

Seems like the classes are called BraidedCategory and SymmetricCategory, so I changed the docstring to reflect this.

@@ -17,9 +17,8 @@ as well as braided functors.
 
 ## Implementation note
 
-We make `BraidedMonoidalCategory` another typeclass, but then have `SymmetricMonoidalCategory`
-extend this. The rationale is that we are not carrying any additional data,
-just requiring a property.
+We make `BraidedCategory` another typeclass, but then have `SymmetricCategory` extend this.
+The rationale is that we are not carrying any additional data, just requiring a property.
 
 ## Future work
 

We remove all possible occurences of Type _ and Sort _ in favor of Type* and Sort*.

This has nice performance benefits.

@@ -75,7 +75,7 @@ notation "β_" => braiding
 Verifying the axioms for a braiding by checking that the candidate braiding is sent to a braiding
 by a faithful monoidal functor.
 -/
-def braidedCategoryOfFaithful {C D : Type _} [Category C] [Category D] [MonoidalCategory C]
+def braidedCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalCategory C]
     [MonoidalCategory D] (F : MonoidalFunctor C D) [Faithful F.toFunctor] [BraidedCategory D]
     (β : ∀ X Y : C, X ⊗ Y ≅ Y ⊗ X)
     (w : ∀ X Y, F.μ _ _ ≫ F.map (β X Y).hom = (β_ _ _).hom ≫ F.μ _ _) : BraidedCategory C where
@@ -111,7 +111,7 @@ def braidedCategoryOfFaithful {C D : Type _} [Category C] [Category D] [Monoidal
 #align category_theory.braided_category_of_faithful CategoryTheory.braidedCategoryOfFaithful
 
 /-- Pull back a braiding along a fully faithful monoidal functor. -/
-noncomputable def braidedCategoryOfFullyFaithful {C D : Type _} [Category C] [Category D]
+noncomputable def braidedCategoryOfFullyFaithful {C D : Type*} [Category C] [Category D]
     [MonoidalCategory C] [MonoidalCategory D] (F : MonoidalFunctor C D) [Full F.toFunctor]
     [Faithful F.toFunctor] [BraidedCategory D] : BraidedCategory C :=
   braidedCategoryOfFaithful F
@@ -308,7 +308,7 @@ attribute [simp] BraidedFunctor.braided
 /--
 A braided category with a faithful braided functor to a symmetric category is itself symmetric.
 -/
-def symmetricCategoryOfFaithful {C D : Type _} [Category C] [Category D] [MonoidalCategory C]
+def symmetricCategoryOfFaithful {C D : Type*} [Category C] [Category D] [MonoidalCategory C]
     [MonoidalCategory D] [BraidedCategory C] [SymmetricCategory D] (F : BraidedFunctor C D)
     [Faithful F.toFunctor] : SymmetricCategory C where
   symmetry X Y := F.map_injective (by simp)

git ls-files '*.lean' | xargs sed -i "s=maxHeartbeats [0-9]*$=& in="

Affected files:

Mathlib/CategoryTheory/Monoidal/Braided.lean:479:set_option maxHeartbeats 400000
Mathlib/FieldTheory/Adjoin.lean:1212:set_option synthInstance.maxHeartbeats 30000
Mathlib/FieldTheory/IsAlgClosed/Basic.lean:302:set_option maxHeartbeats 800000
Mathlib/FieldTheory/IsAlgClosed/Basic.lean:303:set_option synthInstance.maxHeartbeats 400000
Mathlib/RepresentationTheory/GroupCohomology/Basic.lean:202:set_option maxHeartbeats 6400000
Mathlib/RepresentationTheory/GroupCohomology/Resolution.lean:345:set_option maxHeartbeats 800000
Mathlib/Topology/MetricSpace/GromovHausdorff.lean:415:set_option maxHeartbeats 300000

Zulip

@@ -476,7 +476,7 @@ theorem tensor_associativity_aux (W X Y Z : C) :
   slice_rhs 3 5 => rw [← tensor_comp, ← tensor_comp, ← hexagon_reverse, tensor_comp, tensor_comp]
 #align category_theory.tensor_associativity_aux CategoryTheory.tensor_associativity_aux
 
-set_option maxHeartbeats 400000
+set_option maxHeartbeats 400000 in
 theorem tensor_associativity (X₁ X₂ Y₁ Y₂ Z₁ Z₂ : C) :
     (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (Z₁ ⊗ Z₂)) ≫
         tensor_μ C (X₁ ⊗ Y₁, X₂ ⊗ Y₂) (Z₁, Z₂) ≫ ((α_ X₁ Y₁ Z₁).hom ⊗ (α_ X₂ Y₂ Z₂).hom) =
@@ -625,6 +625,7 @@ theorem associator_monoidal_aux (W X Y Z : C) :
   simp
 #align category_theory.associator_monoidal_aux CategoryTheory.associator_monoidal_aux
 
+set_option maxHeartbeats 400000 in
 theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ : C) :
     tensor_μ C (X₁ ⊗ X₂, X₃) (Y₁ ⊗ Y₂, Y₃) ≫
         (tensor_μ C (X₁, X₂) (Y₁, Y₂) ⊗ 𝟙 (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =

• depends on: #5966

Co-authored-by: Eric Wieser <wieser.eric@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@gmail.com>

@@ -2,16 +2,13 @@
 Copyright (c) 2020 Scott Morrison. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Scott Morrison
-
-! This file was ported from Lean 3 source module category_theory.monoidal.braided
-! leanprover-community/mathlib commit 2efd2423f8d25fa57cf7a179f5d8652ab4d0df44
-! Please do not edit these lines, except to modify the commit id
-! if you have ported upstream changes.
 -/
 import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
 import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
 import Mathlib.CategoryTheory.Monoidal.Discrete
 
+#align_import category_theory.monoidal.braided from "leanprover-community/mathlib"@"2efd2423f8d25fa57cf7a179f5d8652ab4d0df44"
+
 /-!
 # Braided and symmetric monoidal categories
 

Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au> Co-authored-by: Joël Riou <37772949+joelriou@users.noreply.github.com>

@@ -273,6 +273,11 @@ instance categoryLaxBraidedFunctor : Category (LaxBraidedFunctor C D) :=
   InducedCategory.category LaxBraidedFunctor.toLaxMonoidalFunctor
 #align category_theory.lax_braided_functor.category_lax_braided_functor CategoryTheory.LaxBraidedFunctor.categoryLaxBraidedFunctor
 
+-- Porting note: added, as `MonoidalNatTrans.ext` does not apply to morphisms.
+@[ext]
+lemma ext' {F G : LaxBraidedFunctor C D} {α β : F ⟶ G} (w : ∀ X : C, α.app X = β.app X) : α = β :=
+  MonoidalNatTrans.ext _ _ (funext w)
+
 @[simp]
 theorem comp_toNatTrans {F G H : LaxBraidedFunctor C D} {α : F ⟶ G} {β : G ⟶ H} :
     (α ≫ β).toNatTrans = @CategoryStruct.comp (C ⥤ D) _ _ _ _ α.toNatTrans β.toNatTrans :=
@@ -342,6 +347,11 @@ instance categoryBraidedFunctor : Category (BraidedFunctor C D) :=
   InducedCategory.category BraidedFunctor.toMonoidalFunctor
 #align category_theory.braided_functor.category_braided_functor CategoryTheory.BraidedFunctor.categoryBraidedFunctor
 
+-- Porting note: added, as `MonoidalNatTrans.ext` does not apply to morphisms.
+@[ext]
+lemma ext' {F G : BraidedFunctor C D} {α β : F ⟶ G} (w : ∀ X : C, α.app X = β.app X) : α = β :=
+  MonoidalNatTrans.ext _ _ (funext w)
+
 @[simp]
 theorem comp_toNatTrans {F G H : BraidedFunctor C D} {α : F ⟶ G} {β : G ⟶ H} :
     (α ≫ β).toNatTrans = @CategoryStruct.comp (C ⥤ D) _ _ _ _ α.toNatTrans β.toNatTrans :=

These are all doc fixes

@@ -279,7 +279,7 @@ theorem comp_toNatTrans {F G H : LaxBraidedFunctor C D} {α : F ⟶ G} {β : G 
   rfl
 #align category_theory.lax_braided_functor.comp_to_nat_trans CategoryTheory.LaxBraidedFunctor.comp_toNatTrans
 
-/-- Interpret a natural isomorphism of the underlyling lax monoidal functors as an
+/-- Interpret a natural isomorphism of the underlying lax monoidal functors as an
 isomorphism of the lax braided monoidal functors.
 -/
 @[simps]

Co-authored-by: Scott Morrison <scott.morrison@gmail.com> Co-authored-by: Scott Morrison <scott.morrison@anu.edu.au>

@@ -32,7 +32,7 @@ just requiring a property.
 -/
 
 
-open CategoryTheory
+open CategoryTheory MonoidalCategory
 
 universe v v₁ v₂ v₃ u u₁ u₂ u₃
 

@@ -294,7 +294,7 @@ end LaxBraidedFunctor
 which preserves the braiding.
 -/
 structure BraidedFunctor extends MonoidalFunctor C D where
-  -- Note this is stated differently than for `lax_braided_functor`.
+  -- Note this is stated differently than for `LaxBraidedFunctor`.
   -- We move the `μ X Y` to the right hand side,
   -- so that this makes a good `@[simp]` lemma.
   braided : ∀ X Y : C, map (β_ X Y).hom = inv (μ X Y) ≫ (β_ (obj X) (obj Y)).hom ≫ μ Y X := by
@@ -308,8 +308,8 @@ A braided category with a faithful braided functor to a symmetric category is it
 -/
 def symmetricCategoryOfFaithful {C D : Type _} [Category C] [Category D] [MonoidalCategory C]
     [MonoidalCategory D] [BraidedCategory C] [SymmetricCategory D] (F : BraidedFunctor C D)
-    [Faithful F.toFunctor] : SymmetricCategory C
-    where symmetry X Y := F.map_injective (by simp)
+    [Faithful F.toFunctor] : SymmetricCategory C where
+  symmetry X Y := F.map_injective (by simp)
 #align category_theory.symmetric_category_of_faithful CategoryTheory.symmetricCategoryOfFaithful
 
 namespace BraidedFunctor
@@ -362,7 +362,8 @@ section CommMonoid
 
 variable (M : Type u) [CommMonoid M]
 
-instance : BraidedCategory (Discrete M) where braiding X Y := Discrete.eqToIso (mul_comm X.as Y.as)
+instance : BraidedCategory (Discrete M) where
+  braiding X Y := Discrete.eqToIso (mul_comm X.as Y.as)
 
 variable {M} {N : Type u} [CommMonoid N]
 

Co-authored-by: Scott Morrison <scott.morrison@gmail.com>