mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| ((α_ R.X P.X S.X).hom ≫ R.X ◁ P.actRight ≫ P.actLeft) ▷ Q.X

pp: case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X (P.X ⊗ S.X) Q.X).inv

pp: case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X P.X S.X).inv ▷ Q.X

Tactic:
comp_whiskerRight

Goal state after:
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X P.X S.X).hom ▷ Q.X ≫ (R.X ◁ P.actRight ≫ P.actLeft) ▷ Q.X

case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X (P.X ⊗ S.X) Q.X).inv

case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X P.X S.X).inv ▷ Q.X
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft ≫ P.actLeft) ▷ Q.X

pp: case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)

pp: case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ (R.X ⊗ R.X) P.X Q.X).inv

Tactic:
comp_whiskerRight

Goal state after:
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ R.X R.X P.X).hom ▷ Q.X ≫ (R.X ◁ P.actLeft) ▷ Q.X ≫ P.actLeft ▷ Q.X

case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)

case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
| (α_ (R.X ⊗ R.X) P.X Q.X).inv
Module: Mathlib.CategoryTheory.Monoidal.Arrow

Goal state before:
pp: C : Type u
inst✝⁵ : Category.{v, u} C
inst✝⁴ : HasPushouts C
inst✝³ : HasInitial C
inst✝² : CartesianMonoidalCategory C
inst✝¹ : MonoidalClosed C
inst✝ : BraidedCategory C
X Y Z : Arrow C
⊢ (α_ X Y Z).inv ≫ (braiding (X ⊗ Y) Z).hom ≫ (α_ Z X Y).inv =
    X ◁ (braiding Y Z).hom ≫ (α_ X Z Y).inv ≫ (braiding X Z).hom ▷ Y

Tactic:
refine Arrow.hom_ext _ _ (pushout.hom_ext ?_ (by simp)) (by simp)
  apply ((tensorLeft _).map_isPushout (IsPushout.of_hasPushout _ _)).hom_ext <;> simp

Goal state after:
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| (α_ P.X Q.X T.X).hom ▷ T.X ≫
    (P.X ◁ Q.actRight ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X

pp: case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| actRight P Q

pp: case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| (α_ (P.X ⊗ Q.X) T.X T.X).inv

Tactic:
comp_whiskerRight

Goal state after:
case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| (α_ P.X Q.X T.X).hom ▷ T.X ≫
    (P.X ◁ Q.actRight) ▷ T.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X

case a.a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| actRight P Q

case a
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| (α_ (P.X ⊗ Q.X) T.X T.X).inv
Module: Mathlib.CategoryTheory.Monoidal.Ring

Goal state before:
pp: case refine_1
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
R : C
inst✝¹ : MonObj R
inst✝ : AddMonObj R
h : R ◁ σ ≫ μ = lift (R ◁ fst R R ≫ μ) (R ◁ snd R R ≫ μ) ≫ σ
x✝ : C
a b c : x✝ ⟶ R
⊢ a * (b + c) = a * b + a * c

pp: case refine_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
R : C
inst✝¹ : MonObj R
inst✝ : AddMonObj R
h : ∀ ⦃X : C⦄ (a b c : X ⟶ R), a * (b + c) = a * b + a * c
⊢ R ◁ σ ≫ μ = lift (R ◁ fst R R ≫ μ) (R ◁ snd R R ≫ μ) ≫ σ

Tactic:
· have := lift a (lift b c) ≫= h
    simp only [lift_whiskerLeft_assoc] at this
    simp only [Hom.add_def, Hom.mul_def, this, ← Category.assoc]
    cat_disch

Goal state after:
case refine_2
C : Type u
inst✝³ : Category.{v, u} C
inst✝² : CartesianMonoidalCategory C
R : C
inst✝¹ : MonObj R
inst✝ : AddMonObj R
h : ∀ ⦃X : C⦄ (a b c : X ⟶ R), a * (b + c) = a * b + a * c
⊢ R ◁ σ ≫ μ = lift (R ◁ fst R R ≫ μ) (R ◁ snd R R ≫ μ) ≫ σ
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| (α_ R.X P.X Q.X).inv ▷ T.X ≫
    (P.actLeft ▷ Q.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) ▷ T.X

pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| actRight P Q

Tactic:
comp_whiskerRight

Goal state after:
case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| (α_ R.X P.X Q.X).inv ▷ T.X ≫
    P.actLeft ▷ Q.X ▷ T.X ≫ coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ T.X

case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
R S T : Mon C
P : Bimod R S
Q : Bimod S T
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
| actRight P Q
Module: Mathlib.CategoryTheory.Monoidal.Closed.FunctorCategory.Basic

Goal state before:
pp: case e_a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalClosed C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₂' F₃ F₃' : J ⥤ C
g : F₂ ⟶ functorEnrichedHom C F₁ F₃
j j' : J
φ : j ⟶ j'
⊢ enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) ((Under.map φ).obj (Under.mk (𝟙 j'))) ≫
      eHomWhiskerRight C (F₁.map φ) (F₃.obj j') =
    enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) (Under.mk (𝟙 j)) ≫
      eHomWhiskerLeft C (F₁.obj j) (F₃.map φ)

Tactic:
let α : Under.mk (𝟙 j) ⟶ (Under.map φ).obj (Under.mk (𝟙 j')) := Under.homMk φ

Goal state after:
case e_a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : MonoidalClosed C
J : Type u₂
inst✝¹ : Category.{v₂, u₂} J
inst✝ : ∀ (F₁ F₂ : J ⥤ C), HasFunctorEnrichedHom C F₁ F₂
F₁ F₂ F₂' F₃ F₃' : J ⥤ C
g : F₂ ⟶ functorEnrichedHom C F₁ F₃
j j' : J
φ : j ⟶ j'
α : Under.mk (𝟙 j) ⟶ (Under.map φ).obj (Under.mk (𝟙 j')) := Under.homMk φ ⋯
⊢ enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) ((Under.map φ).obj (Under.mk (𝟙 j'))) ≫
      eHomWhiskerRight C (F₁.map φ) (F₃.obj j') =
    enrichedHomπ C (Under.forget j ⋙ F₁) (Under.forget j ⋙ F₃) (Under.mk (𝟙 j)) ≫
      eHomWhiskerLeft C (F₁.obj j) (F₃.map φ)
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
| (M.X ◁ f.hom ≫
      colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one) ▷
    Z.X

pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
| TensorBimod.actRight M N₂

Tactic:
comp_whiskerRight

Goal state after:
case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
| (M.X ◁ f.hom) ▷ Z.X ≫
    colimit.ι (parallelPair (M.actRight ▷ N₂.X) ((α_ M.X Y.X N₂.X).hom ≫ M.X ◁ N₂.actLeft)) WalkingParallelPair.one ▷
      Z.X

case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M✝ : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M : Bimod X Y
N₁ N₂ : Bimod Y Z
f : N₁ ⟶ N₂
| TensorBimod.actRight M N₂
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
⊢ (f.hom ▷ Y.X ≫ M₂.actRight) ▷ N.X = f.hom ▷ Y.X ▷ N.X ≫ M₂.actRight ▷ N.X

Tactic:
comp_whiskerRight

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
⊢ f.hom ▷ Y.X ▷ N.X ≫ M₂.actRight ▷ N.X = f.hom ▷ Y.X ▷ N.X ≫ M₂.actRight ▷ N.X
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
| (f.hom ▷ N.X ≫
      colimit.ι (parallelPair (M₂.actRight ▷ N.X) ((α_ M₂.X Y.X N.X).hom ≫ M₂.X ◁ N.actLeft)) WalkingParallelPair.one) ▷
    Z.X

pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
| TensorBimod.actRight M₂ N

Tactic:
comp_whiskerRight

Goal state after:
case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
| f.hom ▷ N.X ▷ Z.X ≫
    colimit.ι (parallelPair (M₂.actRight ▷ N.X) ((α_ M₂.X Y.X N.X).hom ≫ M₂.X ◁ N.actLeft)) WalkingParallelPair.one ▷
      Z.X

case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y Z : Mon C
M₁ M₂ : Bimod X Y
f : M₁ ⟶ M₂
N : Bimod Y Z
| TensorBimod.actRight M₂ N
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ≫
      coequalizer.desc
        ((α_ P.X Q.X L.X).hom ≫
          P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
            coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
              ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
        ⋯) ▷
    U.X

pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| TensorBimod.actRight P (Q.tensorBimod L)

Tactic:
comp_whiskerRight

Goal state after:
case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)).inv ▷ U.X ≫
    coequalizer.desc
        ((α_ P.X Q.X L.X).hom ≫
          P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
            coequalizer.π (P.actRight ▷ TensorBimod.X Q L)
              ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ P.X ◁ TensorBimod.actLeft Q L))
        ⋯ ▷
      U.X

case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| TensorBimod.actRight P (Q.tensorBimod L)
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (α_ P.X Q.X L.X).hom ▷ U.X ≫
    (P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
        coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
          ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
            P.X ◁ TensorBimod.actLeft Q L)) ▷
      U.X

pp: case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| TensorBimod.actRight P (Q.tensorBimod L)

Tactic:
comp_whiskerRight

Goal state after:
case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (α_ P.X Q.X L.X).hom ▷ U.X ≫
    (P.X ◁ coequalizer.π (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft)) ▷ U.X ≫
      coequalizer.π (P.actRight ▷ coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))
          ((α_ P.X S.X (coequalizer (Q.actRight ▷ L.X) ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft))).hom ≫
            P.X ◁ TensorBimod.actLeft Q L) ▷
        U.X

case a
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| TensorBimod.actRight P (Q.tensorBimod L)
Module: Mathlib.CategoryTheory.Monoidal.Bimod

Goal state before:
pp: C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (((α_ P.X (Q.X ⊗ T.X) L.X).inv ≫ ((α_ P.X Q.X T.X).inv ≫ (α_ P.X Q.X T.X).hom ≫ P.X ◁ Q.actRight) ▷ L.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X) ≫
      coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
        ((α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X L.X).hom ≫
          coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) =
    P.X ◁ ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
      (α_ P.X Q.X L.X).inv ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
          coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
            ((α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X L.X).hom ≫
              coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)

Tactic:
comp_whiskerRight

Goal state after:
C : Type u₁
inst✝⁴ : Category.{v₁, u₁} C
inst✝³ : MonoidalCategory C
A B : Mon C
M : Bimod A B
inst✝² : HasCoequalizers C
inst✝¹ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst✝ : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
R S T U : Mon C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
⊢ (((α_ P.X (Q.X ⊗ T.X) L.X).inv ≫ (α_ P.X Q.X T.X).inv ▷ L.X ≫ ((α_ P.X Q.X T.X).hom ≫ P.X ◁ Q.actRight) ▷ L.X) ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X) ≫
      coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
        ((α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X L.X).hom ≫
          coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft) =
    P.X ◁ ((α_ Q.X T.X L.X).hom ≫ Q.X ◁ L.actLeft) ≫
      (α_ P.X Q.X L.X).inv ≫
        coequalizer.π (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ▷ L.X ≫
          coequalizer.π (TensorBimod.actRight P Q ▷ L.X)
            ((α_ (coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft)) T.X L.X).hom ≫
              coequalizer (P.actRight ▷ Q.X) ((α_ P.X S.X Q.X).hom ≫ P.X ◁ Q.actLeft) ◁ L.actLeft)
Module: Mathlib.CategoryTheory.Monoidal.Hopf_

Goal state before:
pp: C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
⊢ (Δ ⊗ₘ Δ) ≫
      (α_ A A (A ⊗ A)).hom ≫
        A ◁ (α_ A A A).inv ≫
          A ◁ (β_ A A).hom ▷ A ≫
            (α_ A (A ⊗ A) A).inv ≫ (α_ A A A).inv ▷ A ≫ μ ▷ A ▷ A ≫ 𝒮 ▷ A ▷ A ≫ (α_ A A A).hom ≫ A ◁ μ ≫ μ =
    (ε ⊗ₘ ε) ≫ (λ_ (𝟙_ C)).hom ≫ η

Tactic:
slice_lhs 8 9 =>
    rw [associator_naturality_left]

Goal state after:
C : Type u₁
inst✝³ : Category.{v₁, u₁} C
inst✝² : MonoidalCategory C
inst✝¹ : BraidedCategory C
A : C
inst✝ : HopfObj A
⊢ (Δ ⊗ₘ Δ) ≫
      (α_ A A (A ⊗ A)).hom ≫
        A ◁ (α_ A A A).inv ≫
          A ◁ (β_ A A).hom ▷ A ≫
            (α_ A (A ⊗ A) A).inv ≫ (α_ A A A).inv ▷ A ≫ μ ▷ A ▷ A ≫ (((α_ A A A).hom ≫ 𝒮 ▷ (A ⊗ A)) ≫ A ◁ μ) ≫ μ =
    (ε ⊗ₘ ε) ≫ (λ_ (𝟙_ C)).hom ≫ η