l3lab/ntp-mathlib huggingface dataset:
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => 

Proof state:
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} (tensorRight X)
| (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft) ⊗ 𝟙 T.X
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} (tensorRight X)
| (α_ P.X Q.X T.X).hom
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} (tensorRight X)
| 𝟙 P.X ⊗ Q.actRight

Next tactic:
rw [comp_tensor_id]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          

Proof state:
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} (tensorRight X)
⊢ ((((α_ P.X S.X Q.X).hom ⊗ 𝟙 T.X) ≫ ((𝟙 P.X ⊗ Q.actLeft) ⊗ 𝟙 T.X)) ≫ (α_ P.X Q.X T.X).hom) ≫ (𝟙 P.X ⊗ Q.actRight) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_lhs 2 3 => rw [associator_naturality]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          slice_lhs 2 3 => 

Proof state:
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} (tensorRight X)
| ((𝟙 P.X ⊗ Q.actLeft) ⊗ 𝟙 T.X) ≫ (α_ P.X Q.X T.X).hom
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} (tensorRight X)
| 𝟙 P.X ⊗ Q.actRight
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} (tensorRight X)
| (α_ P.X S.X Q.X).hom ⊗ 𝟙 T.X

Next tactic:
rw [associator_naturality]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          slice_lhs 2 3 => rw [associator_naturality]
          

Proof state:
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} (tensorRight X)
⊢ ((α_ P.X S.X Q.X).hom ⊗ 𝟙 T.X) ≫ ((α_ P.X (S.X ⊗ Q.X) T.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft ⊗ 𝟙 T.X)) ≫ (𝟙 P.X ⊗ Q.actRight) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_lhs 3 4 => rw [← id_tensor_comp, middle_assoc, id_tensor_comp]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          slice_lhs 2 3 => rw [associator_naturality]
          slice_lhs 3 4 => 

Proof state:
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} (tensorRight X)
| (𝟙 P.X ⊗ Q.actLeft ⊗ 𝟙 T.X) ≫ (𝟙 P.X ⊗ Q.actRight)
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} (tensorRight X)
| (α_ P.X S.X Q.X).hom ⊗ 𝟙 T.X
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} (tensorRight X)
| (α_ P.X (S.X ⊗ Q.X) T.X).hom

Next tactic:
rw [← id_tensor_comp, middle_assoc, id_tensor_comp]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          slice_lhs 2 3 => rw [associator_naturality]
          slice_lhs 3 4 => rw [← id_tensor_comp, middle_assoc, id_tensor_comp]
          

Proof state:
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} (tensorRight X)
⊢ ((α_ P.X S.X Q.X).hom ⊗ 𝟙 T.X) ≫
      (α_ P.X (S.X ⊗ Q.X) T.X).hom ≫ (𝟙 P.X ⊗ (α_ S.X Q.X T.X).hom) ≫ (𝟙 P.X ⊗ (𝟙 S.X ⊗ Q.actRight) ≫ Q.actLeft) =
    ((α_ (P.X ⊗ S.X) Q.X T.X).hom ≫
        (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ P.X S.X Q.X).inv) ≫
      (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_rhs 4 6 => rw [Iso.inv_hom_id_assoc]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          slice_lhs 2 3 => rw [associator_naturality]
          slice_lhs 3 4 => rw [← id_tensor_comp, middle_assoc, id_tensor_comp]
          slice_rhs 4 6 => 

Proof state:
case a.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} (tensorRight X)
| (α_ P.X S.X Q.X).inv ≫ (α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)
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} (tensorRight X)
| (α_ (P.X ⊗ S.X) Q.X T.X).hom
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} (tensorRight X)
| (α_ P.X S.X (Q.X ⊗ T.X)).hom
case a.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} (tensorRight X)
| 𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight

Next tactic:
rw [Iso.inv_hom_id_assoc]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          slice_lhs 2 3 => rw [associator_naturality]
          slice_lhs 3 4 => rw [← id_tensor_comp, middle_assoc, id_tensor_comp]
          slice_rhs 4 6 => rw [Iso.inv_hom_id_assoc]
          

Proof state:
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} (tensorRight X)
⊢ ((α_ P.X S.X Q.X).hom ⊗ 𝟙 T.X) ≫
      (α_ P.X (S.X ⊗ Q.X) T.X).hom ≫ (𝟙 P.X ⊗ (α_ S.X Q.X T.X).hom) ≫ (𝟙 P.X ⊗ (𝟙 S.X ⊗ Q.actRight) ≫ Q.actLeft) =
    (α_ (P.X ⊗ S.X) Q.X T.X).hom ≫ (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (𝟙 P.X ⊗ Q.actLeft)

Next tactic:
slice_rhs 3 4 => rw [← id_tensor_comp]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          slice_lhs 2 3 => rw [associator_naturality]
          slice_lhs 3 4 => rw [← id_tensor_comp, middle_assoc, id_tensor_comp]
          slice_rhs 4 6 => rw [Iso.inv_hom_id_assoc]
          slice_rhs 3 4 => 

Proof state:
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} (tensorRight X)
| (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (𝟙 P.X ⊗ Q.actLeft)
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} (tensorRight X)
| (α_ (P.X ⊗ S.X) Q.X T.X).hom
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} (tensorRight X)
| (α_ P.X S.X (Q.X ⊗ T.X)).hom

Next tactic:
rw [← id_tensor_comp]
/-- Right action for the tensor product of two bimodules. -/
noncomputable def actRight : X P Q ⊗ T.X ⟶ X P Q :=
  (PreservesCoequalizer.iso (tensorRight T.X) _ _).inv ≫
    colimMap
      (parallelPairHom _ _ _ _
        ((α_ _ _ _).hom ≫ (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ 𝟙 S.X ⊗ Q.actRight) ≫ (α_ _ _ _).inv)
        ((α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight))
        (by
          dsimp
          slice_lhs 1 2 => rw [associator_naturality]
          slice_lhs 2 3 => rw [tensor_id, tensor_id_comp_id_tensor]
          slice_rhs 3 4 => rw [associator_inv_naturality]
          slice_rhs 2 4 => rw [Iso.hom_inv_id_assoc]
          slice_rhs 2 3 => rw [tensor_id, id_tensor_comp_tensor_id])
        (by
          dsimp
          slice_lhs 1 1 => rw [comp_tensor_id]
          slice_lhs 2 3 => rw [associator_naturality]
          slice_lhs 3 4 => rw [← id_tensor_comp, middle_assoc, id_tensor_comp]
          slice_rhs 4 6 => rw [Iso.inv_hom_id_assoc]
          slice_rhs 3 4 => rw [← id_tensor_comp]
          

Proof state:
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} (tensorRight X)
⊢ ((α_ P.X S.X Q.X).hom ⊗ 𝟙 T.X) ≫
      (α_ P.X (S.X ⊗ Q.X) T.X).hom ≫ (𝟙 P.X ⊗ (α_ S.X Q.X T.X).hom) ≫ (𝟙 P.X ⊗ (𝟙 S.X ⊗ Q.actRight) ≫ Q.actLeft) =
    (α_ (P.X ⊗ S.X) Q.X T.X).hom ≫ (α_ P.X S.X (Q.X ⊗ T.X)).hom ≫ (𝟙 P.X ⊗ (𝟙 S.X ⊗ Q.actRight) ≫ Q.actLeft)

Next tactic:
coherence
theorem π_tensor_id_actRight :
    (coequalizer.π _ _ ⊗ 𝟙 T.X) ≫ actRight P Q =
      (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight) ≫ coequalizer.π _ _ := by
  

Proof state:
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} (tensorRight X)
⊢ (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 T.X) ≫ actRight P Q =
    (α_ P.X Q.X T.X).hom ≫
      (𝟙 P.X ⊗ Q.actRight) ≫ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))

Next tactic:
erw [map_π_preserves_coequalizer_inv_colimMap (tensorRight _)]
theorem π_tensor_id_actRight :
    (coequalizer.π _ _ ⊗ 𝟙 T.X) ≫ actRight P Q =
      (α_ _ _ _).hom ≫ (𝟙 P.X ⊗ Q.actRight) ≫ coequalizer.π _ _ := by
  erw [map_π_preserves_coequalizer_inv_colimMap (tensorRight _)]
  

Proof state:
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} (tensorRight X)
⊢ ((α_ P.X Q.X T.X).hom ≫ (𝟙 P.X ⊗ Q.actRight)) ≫
      coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) =
    (α_ P.X Q.X T.X).hom ≫
      (𝟙 P.X ⊗ Q.actRight) ≫ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))

Next tactic:
simp only [Category.assoc]
theorem actRight_one' : (𝟙 _ ⊗ T.one) ≫ actRight P Q = (ρ_ _).hom := by
  

Proof state:
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} (tensorRight X)
⊢ (𝟙 (X P Q) ⊗ T.one) ≫ actRight P Q = (ρ_ (X P Q)).hom

Next tactic:
refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
theorem actRight_one' : (𝟙 _ ⊗ T.one) ≫ actRight P Q = (ρ_ _).hom := by
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  

Proof state:
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} (tensorRight X)
⊢ (tensorRight (𝟙_ C)).map (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (𝟙 (X P Q) ⊗ T.one) ≫ actRight P Q =
    (tensorRight (𝟙_ C)).map (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ≫
      (ρ_ (X P Q)).hom

Next tactic:
dsimp [X]
theorem actRight_one' : (𝟙 _ ⊗ T.one) ≫ actRight P Q = (ρ_ _).hom := by
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  

Proof state:
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} (tensorRight X)
⊢ (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 (𝟙_ C)) ≫
      (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ T.one) ≫ actRight P Q =
    (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 (𝟙_ C)) ≫
      (ρ_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom

Next tactic:
slice_lhs 1 2 => erw [tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
theorem actRight_one' : (𝟙 _ ⊗ T.one) ≫ actRight P Q = (ρ_ _).hom := by
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  slice_lhs 1 2 => 

Proof state:
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} (tensorRight X)
| (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 (𝟙_ C)) ≫
    (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ T.one)
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} (tensorRight X)
| actRight P Q

Next tactic:
erw [tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
theorem actRight_one' : (𝟙 _ ⊗ T.one) ≫ actRight P Q = (ρ_ _).hom := by
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  slice_lhs 1 2 => erw [tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
  

Proof state:
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} (tensorRight X)
⊢ ((𝟙 (P.X ⊗ Q.X) ⊗ T.one) ≫
        (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 T.X)) ≫
      actRight P Q =
    (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 (𝟙_ C)) ≫
      (ρ_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom

Next tactic:
slice_lhs 2 3 => rw [π_tensor_id_actRight]
theorem actRight_one' : (𝟙 _ ⊗ T.one) ≫ actRight P Q = (ρ_ _).hom := by
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  slice_lhs 1 2 => erw [tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
  slice_lhs 2 3 => 

Proof state:
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} (tensorRight X)
| (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 T.X) ≫ actRight P Q
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} (tensorRight X)
| 𝟙 (P.X ⊗ Q.X) ⊗ T.one

Next tactic:
rw [π_tensor_id_actRight]
theorem actRight_one' : (𝟙 _ ⊗ T.one) ≫ actRight P Q = (ρ_ _).hom := by
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp [X]
  -- porting note: had to replace `rw` by `erw`
  slice_lhs 1 2 => erw [tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
  slice_lhs 2 3 => rw [π_tensor_id_actRight]
  

Proof state:
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} (tensorRight X)
⊢ (𝟙 (P.X ⊗ Q.X) ⊗ T.one) ≫
      (α_ P.X Q.X T.X).hom ≫
        (𝟙 P.X ⊗ Q.actRight) ≫ coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) =
    (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 (𝟙_ C)) ≫
      (ρ_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)))).hom

Next tactic:
slice_lhs 1 2 => rw [← MonoidalCategory.tensor_id, associator_naturality]