l3lab/ntp-mathlib huggingface dataset:
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => 

Proof state:
case a.a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (TensorBimod.actRight P Q ⊗ 𝟙 L.X) ≫
    coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
      ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
        (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))
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)
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
case a.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)
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).inv ⊗ 𝟙 L.X
case a.a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 T.X) ⊗ 𝟙 L.X

Next tactic:
rw [coequalizer.condition]
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => rw [coequalizer.condition]
        

Proof state:
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 ⊗ 𝟙 L.X) ≫
        ((coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 T.X) ⊗ 𝟙 L.X) ≫
          ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
              (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft)) ≫
            coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
              ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
                (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft)) =
    (𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)) ≫
      (α_ P.X Q.X L.X).inv ≫
        (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X) ≫
          coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
            ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
              (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))

Next tactic:
slice_lhs 3 4 => rw [associator_naturality]
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => rw [coequalizer.condition]
        slice_lhs 3 4 => rw [associator_naturality]
        

Proof state:
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 ⊗ 𝟙 L.X) ≫
        (((α_ (P.X ⊗ Q.X) T.X L.X).hom ≫
              (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 T.X ⊗ 𝟙 L.X)) ≫
            (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft)) ≫
          coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
            ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
              (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft)) =
    (𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)) ≫
      (α_ P.X Q.X L.X).inv ≫
        (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X) ≫
          coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
            ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
              (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))

Next tactic:
slice_lhs 4 5 => rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor]
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => rw [coequalizer.condition]
        slice_lhs 3 4 => rw [associator_naturality]
        slice_lhs 4 5 => rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor]
        

Proof state:
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 ⊗ 𝟙 L.X) ≫
        (α_ (P.X ⊗ Q.X) T.X L.X).hom ≫
          (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ L.actLeft) ≫
            coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
              ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
                (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft)) =
    (𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)) ≫
      (α_ P.X Q.X L.X).inv ≫
        (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X) ≫
          coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
            ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
              (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))

Next tactic:
slice_rhs 1 2 => rw [id_tensor_comp]
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => rw [coequalizer.condition]
        slice_lhs 3 4 => rw [associator_naturality]
        slice_lhs 4 5 => rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor]
        slice_rhs 1 2 => 

Proof state:
case a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)) ≫ (α_ P.X Q.X L.X).inv
case a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
    ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
      (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))

Next tactic:
rw [id_tensor_comp]
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => rw [coequalizer.condition]
        slice_lhs 3 4 => rw [associator_naturality]
        slice_lhs 4 5 => rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor]
        slice_rhs 1 2 => rw [id_tensor_comp]
        

Proof state:
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 ⊗ 𝟙 L.X) ≫
        (α_ (P.X ⊗ Q.X) T.X L.X).hom ≫
          (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ L.actLeft) ≫
            coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
              ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
                (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft)) =
    ((((𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom) ≫ (𝟙 P.X ⊗ 𝟙 Q.X ⊗ L.actLeft)) ≫ (α_ P.X Q.X L.X).inv) ≫
        (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X)) ≫
      coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
        ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
          (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))

Next tactic:
slice_rhs 2 3 => rw [associator_inv_naturality]
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => rw [coequalizer.condition]
        slice_lhs 3 4 => rw [associator_naturality]
        slice_lhs 4 5 => rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor]
        slice_rhs 1 2 => rw [id_tensor_comp]
        slice_rhs 2 3 => 

Proof state:
case a.a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| (𝟙 P.X ⊗ 𝟙 Q.X ⊗ L.actLeft) ≫ (α_ P.X Q.X L.X).inv
case a.a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X
case a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
    ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
      (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))
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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| 𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom

Next tactic:
rw [associator_inv_naturality]
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => rw [coequalizer.condition]
        slice_lhs 3 4 => rw [associator_naturality]
        slice_lhs 4 5 => rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor]
        slice_rhs 1 2 => rw [id_tensor_comp]
        slice_rhs 2 3 => rw [associator_inv_naturality]
        

Proof state:
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 ⊗ 𝟙 L.X) ≫
        (α_ (P.X ⊗ Q.X) T.X L.X).hom ≫
          (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ L.actLeft) ≫
            coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
              ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
                (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft)) =
    (𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom) ≫
      (((α_ P.X Q.X (T.X ⊗ L.X)).inv ≫ ((𝟙 P.X ⊗ 𝟙 Q.X) ⊗ L.actLeft)) ≫
          (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X)) ≫
        coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
          ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
            (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))

Next tactic:
slice_rhs 3 4 => rw [MonoidalCategory.tensor_id, id_tensor_comp_tensor_id]
/-- An auxiliary morphism for the definition of the underlying morphism of the inverse component of
the associator isomorphism. -/
noncomputable def invAux : P.X ⊗ (Q.tensorBimod L).X ⟶ ((P.tensorBimod Q).tensorBimod L).X :=
  (PreservesCoequalizer.iso (tensorLeft P.X) _ _).inv ≫
    coequalizer.desc ((α_ _ _ _).inv ≫ (coequalizer.π _ _ ⊗ 𝟙 L.X) ≫ coequalizer.π _ _)
      (by
        dsimp; dsimp [TensorBimod.X]
        slice_lhs 1 2 => rw [associator_inv_naturality]
        rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), comp_tensor_id]
        slice_lhs 3 4 =>
          rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            comp_tensor_id]
        slice_lhs 4 5 => rw [coequalizer.condition]
        slice_lhs 3 4 => rw [associator_naturality]
        slice_lhs 4 5 => rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor]
        slice_rhs 1 2 => rw [id_tensor_comp]
        slice_rhs 2 3 => rw [associator_inv_naturality]
        slice_rhs 3 4 => 

Proof state:
case a.a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| ((𝟙 P.X ⊗ 𝟙 Q.X) ⊗ L.actLeft) ≫
    (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X)
case a.a.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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| coequalizer.π (TensorBimod.actRight P Q ⊗ 𝟙 L.X)
    ((α_ (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) T.X L.X).hom ≫
      (𝟙 (coequalizer (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))) ⊗ L.actLeft))
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)
R S T U : Mon_ C
P : Bimod R S
Q : Bimod S T
L : Bimod T U
| 𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom
case a.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)
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

Next tactic:
rw [MonoidalCategory.tensor_id, id_tensor_comp_tensor_id]