l3lab/ntp-mathlib huggingface dataset:
theorem hom_right_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actRight ≫ hom P Q L =
      (hom P Q L ⊗ 𝟙 U.X) ≫ (P.tensorBimod (Q.tensorBimod L)).actRight := by
  dsimp; dsimp [hom, homAux]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  rw [tensorRight_map]
  slice_lhs 1 2 => rw [TensorBimod.π_tensor_id_actRight]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => rw [← comp_tensor_id, coequalizer.π_desc, comp_tensor_id]
  refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp; dsimp [TensorBimod.X]
  slice_lhs 1 2 => rw [associator_naturality]
  slice_lhs 2 3 =>
    rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
  slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_lhs 2 3 => rw [← MonoidalCategory.tensor_id, associator_naturality]
  slice_rhs 1 3 =>
    rw [← comp_tensor_id, ← comp_tensor_id, π_tensor_id_preserves_coequalizer_inv_desc,
      comp_tensor_id, comp_tensor_id]
  slice_rhs 3 4 => erw [TensorBimod.π_tensor_id_actRight P (Q.tensorBimod L)]
  slice_rhs 2 3 => erw [associator_naturality]
  dsimp
  

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

Next tactic:
slice_rhs 3 4 =>
    rw [← id_tensor_comp, TensorBimod.π_tensor_id_actRight, id_tensor_comp, id_tensor_comp]
theorem hom_right_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actRight ≫ hom P Q L =
      (hom P Q L ⊗ 𝟙 U.X) ≫ (P.tensorBimod (Q.tensorBimod L)).actRight := by
  dsimp; dsimp [hom, homAux]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  rw [tensorRight_map]
  slice_lhs 1 2 => rw [TensorBimod.π_tensor_id_actRight]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => rw [← comp_tensor_id, coequalizer.π_desc, comp_tensor_id]
  refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp; dsimp [TensorBimod.X]
  slice_lhs 1 2 => rw [associator_naturality]
  slice_lhs 2 3 =>
    rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
  slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_lhs 2 3 => rw [← MonoidalCategory.tensor_id, associator_naturality]
  slice_rhs 1 3 =>
    rw [← comp_tensor_id, ← comp_tensor_id, π_tensor_id_preserves_coequalizer_inv_desc,
      comp_tensor_id, comp_tensor_id]
  slice_rhs 3 4 => erw [TensorBimod.π_tensor_id_actRight P (Q.tensorBimod L)]
  slice_rhs 2 3 => erw [associator_naturality]
  dsimp
  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 ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)) ⊗ 𝟙 U.X) ≫
    (𝟙 P.X ⊗ TensorBimod.actRight Q L)
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 (TensorBimod.X Q L))
    ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ (𝟙 P.X ⊗ TensorBimod.actLeft Q L))
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 L.X).hom ⊗ 𝟙 U.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
| (α_ P.X (Q.X ⊗ L.X) U.X).hom

Next tactic:
rw [← id_tensor_comp, TensorBimod.π_tensor_id_actRight, id_tensor_comp, id_tensor_comp]
theorem hom_right_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actRight ≫ hom P Q L =
      (hom P Q L ⊗ 𝟙 U.X) ≫ (P.tensorBimod (Q.tensorBimod L)).actRight := by
  dsimp; dsimp [hom, homAux]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  rw [tensorRight_map]
  slice_lhs 1 2 => rw [TensorBimod.π_tensor_id_actRight]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => rw [← comp_tensor_id, coequalizer.π_desc, comp_tensor_id]
  refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
  dsimp; dsimp [TensorBimod.X]
  slice_lhs 1 2 => rw [associator_naturality]
  slice_lhs 2 3 =>
    rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
  slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_lhs 2 3 => rw [← MonoidalCategory.tensor_id, associator_naturality]
  slice_rhs 1 3 =>
    rw [← comp_tensor_id, ← comp_tensor_id, π_tensor_id_preserves_coequalizer_inv_desc,
      comp_tensor_id, comp_tensor_id]
  slice_rhs 3 4 => erw [TensorBimod.π_tensor_id_actRight P (Q.tensorBimod L)]
  slice_rhs 2 3 => erw [associator_naturality]
  dsimp
  slice_rhs 3 4 =>
    rw [← id_tensor_comp, TensorBimod.π_tensor_id_actRight, id_tensor_comp, 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) L.X U.X).hom ≫
      (((α_ P.X Q.X (L.X ⊗ U.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 Q.X ⊗ L.actRight)) ≫
          (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)))) ≫
        coequalizer.π (P.actRight ⊗ 𝟙 (coequalizer (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))))
          ((α_ P.X S.X (coequalizer (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)))).hom ≫
            (𝟙 P.X ⊗ TensorBimod.actLeft Q L)) =
    ((α_ P.X Q.X L.X).hom ⊗ 𝟙 U.X) ≫
      (α_ P.X (Q.X ⊗ L.X) U.X).hom ≫
        ((𝟙 P.X ⊗ (α_ Q.X L.X U.X).hom) ≫
            (𝟙 P.X ⊗ 𝟙 Q.X ⊗ L.actRight) ≫
              (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)))) ≫
          coequalizer.π (P.actRight ⊗ 𝟙 (TensorBimod.X Q L))
            ((α_ P.X S.X (TensorBimod.X Q L)).hom ≫ (𝟙 P.X ⊗ TensorBimod.actLeft Q L))

Next tactic:
coherence
/-- 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
        

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
⊢ (tensorLeft P.X).map (Q.actRight ⊗ 𝟙 L.X) ≫
      (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 P Q).actRight ⊗ 𝟙 L.X)
            ((α_ (tensorBimod P Q).X T.X L.X).hom ≫ (𝟙 (tensorBimod P Q).X ⊗ L.actLeft)) =
    (tensorLeft P.X).map ((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 P Q).actRight ⊗ 𝟙 L.X)
            ((α_ (tensorBimod P Q).X T.X L.X).hom ≫ (𝟙 (tensorBimod P Q).X ⊗ L.actLeft))

Next tactic:
dsimp
/-- 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; 

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
⊢ (𝟙 P.X ⊗ Q.actRight ⊗ 𝟙 L.X) ≫
      (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_ (TensorBimod.X P Q) T.X L.X).hom ≫ (𝟙 (TensorBimod.X P Q) ⊗ 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_ (TensorBimod.X P Q) T.X L.X).hom ≫ (𝟙 (TensorBimod.X P Q) ⊗ L.actLeft))

Next tactic:
dsimp [TensorBimod.X]
/-- 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]
        

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
⊢ (𝟙 P.X ⊗ Q.actRight ⊗ 𝟙 L.X) ≫
      (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)) =
    (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 1 2 => 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 => 

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 ⊗ Q.actRight ⊗ 𝟙 L.X) ≫ (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 [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]
        

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 ≫ ((𝟙 P.X ⊗ Q.actRight) ⊗ 𝟙 L.X)) ≫
        (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)) =
    (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:
rw [← Iso.inv_hom_id_assoc (α_ _ _ _) (𝟙 P.X ⊗ Q.actRight), 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]
        

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).hom ≫ (𝟙 P.X ⊗ Q.actRight) ⊗ 𝟙 L.X)) ≫
        (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)) =
    (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 [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            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 =>
          

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
| ((α_ P.X Q.X T.X).hom ≫ (𝟙 P.X ⊗ Q.actRight) ⊗ 𝟙 L.X) ≫
    (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
| (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

Next tactic:
rw [← comp_tensor_id, Category.assoc, ← TensorBimod.π_tensor_id_actRight,
            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]
        

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) ≫
            (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)) =
    (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 [coequalizer.condition]