l3lab/ntp-mathlib huggingface dataset:
/-- The underlying morphism of the forward component of the associator isomorphism. -/
noncomputable def hom :
    ((P.tensorBimod Q).tensorBimod L).X ⟶ (P.tensorBimod (Q.tensorBimod L)).X :=
  coequalizer.desc (homAux P Q L)
    (by
      dsimp [homAux]
      refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
      dsimp [TensorBimod.X]
      slice_lhs 1 2 =>
        rw [← comp_tensor_id, TensorBimod.π_tensor_id_actRight, comp_tensor_id, comp_tensor_id]
      slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
      slice_lhs 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 ⊗ Q.actRight) ⊗ 𝟙 L.X) ≫ (α_ P.X Q.X L.X).hom
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))
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 (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))
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).hom ⊗ 𝟙 L.X

Next tactic:
rw [associator_naturality]
/-- The underlying morphism of the forward component of the associator isomorphism. -/
noncomputable def hom :
    ((P.tensorBimod Q).tensorBimod L).X ⟶ (P.tensorBimod (Q.tensorBimod L)).X :=
  coequalizer.desc (homAux P Q L)
    (by
      dsimp [homAux]
      refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
      dsimp [TensorBimod.X]
      slice_lhs 1 2 =>
        rw [← comp_tensor_id, TensorBimod.π_tensor_id_actRight, comp_tensor_id, comp_tensor_id]
      slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
      slice_lhs 2 3 => rw [associator_naturality]
      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
| (𝟙 P.X ⊗ Q.actRight ⊗ 𝟙 L.X) ≫
    (𝟙 P.X ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft)))
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 (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))
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).hom ⊗ 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
| (α_ P.X (Q.X ⊗ T.X) L.X).hom

Next tactic:
rw [← id_tensor_comp, coequalizer.condition, id_tensor_comp, id_tensor_comp]
/-- The underlying morphism of the forward component of the associator isomorphism. -/
noncomputable def hom :
    ((P.tensorBimod Q).tensorBimod L).X ⟶ (P.tensorBimod (Q.tensorBimod L)).X :=
  coequalizer.desc (homAux P Q L)
    (by
      dsimp [homAux]
      refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
      dsimp [TensorBimod.X]
      slice_lhs 1 2 =>
        rw [← comp_tensor_id, TensorBimod.π_tensor_id_actRight, comp_tensor_id, comp_tensor_id]
      slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
      slice_lhs 2 3 => rw [associator_naturality]
      slice_lhs 3 4 => rw [← id_tensor_comp, coequalizer.condition, id_tensor_comp, id_tensor_comp]
      slice_rhs 1 2 => rw [associator_naturality]
      slice_rhs 2 3 =>
        rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
      slice_rhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
      

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

Next tactic:
slice_rhs 2 3 => rw [← MonoidalCategory.tensor_id, associator_naturality]
/-- The underlying morphism of the forward component of the associator isomorphism. -/
noncomputable def hom :
    ((P.tensorBimod Q).tensorBimod L).X ⟶ (P.tensorBimod (Q.tensorBimod L)).X :=
  coequalizer.desc (homAux P Q L)
    (by
      dsimp [homAux]
      refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
      dsimp [TensorBimod.X]
      slice_lhs 1 2 =>
        rw [← comp_tensor_id, TensorBimod.π_tensor_id_actRight, comp_tensor_id, comp_tensor_id]
      slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
      slice_lhs 2 3 => rw [associator_naturality]
      slice_lhs 3 4 => rw [← id_tensor_comp, coequalizer.condition, id_tensor_comp, id_tensor_comp]
      slice_rhs 1 2 => rw [associator_naturality]
      slice_rhs 2 3 =>
        rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
      slice_rhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
      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 ⊗ Q.X) ⊗ L.actLeft) ≫ (α_ P.X Q.X L.X).hom
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))
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 (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))
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).hom

Next tactic:
rw [← MonoidalCategory.tensor_id, associator_naturality]
/-- The underlying morphism of the forward component of the associator isomorphism. -/
noncomputable def hom :
    ((P.tensorBimod Q).tensorBimod L).X ⟶ (P.tensorBimod (Q.tensorBimod L)).X :=
  coequalizer.desc (homAux P Q L)
    (by
      dsimp [homAux]
      refine' (cancel_epi ((tensorRight _ ⋙ tensorRight _).map (coequalizer.π _ _))).1 _
      dsimp [TensorBimod.X]
      slice_lhs 1 2 =>
        rw [← comp_tensor_id, TensorBimod.π_tensor_id_actRight, comp_tensor_id, comp_tensor_id]
      slice_lhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
      slice_lhs 2 3 => rw [associator_naturality]
      slice_lhs 3 4 => rw [← id_tensor_comp, coequalizer.condition, id_tensor_comp, id_tensor_comp]
      slice_rhs 1 2 => rw [associator_naturality]
      slice_rhs 2 3 =>
        rw [MonoidalCategory.tensor_id, tensor_id_comp_id_tensor, ← id_tensor_comp_tensor_id]
      slice_rhs 3 5 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
      slice_rhs 2 3 => rw [← MonoidalCategory.tensor_id, 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).hom ⊗ 𝟙 L.X) ≫
      (α_ P.X (Q.X ⊗ T.X) L.X).hom ≫
        (((𝟙 P.X ⊗ (α_ Q.X T.X L.X).hom) ≫ (𝟙 P.X ⊗ 𝟙 Q.X ⊗ L.actLeft)) ≫
            (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) T.X L.X).hom ≫
      (((α_ P.X Q.X (T.X ⊗ L.X)).hom ≫ (𝟙 P.X ⊗ 𝟙 Q.X ⊗ L.actLeft)) ≫
          (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))

Next tactic:
coherence
theorem hom_left_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actLeft ≫ hom P Q L =
      (𝟙 R.X ⊗ hom P Q L) ≫ (P.tensorBimod (Q.tensorBimod L)).actLeft := 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
⊢ (tensorBimod (tensorBimod P Q) L).actLeft ≫ hom P Q L =
    (𝟙 R.X ⊗ hom P Q L) ≫ (tensorBimod P (tensorBimod Q L)).actLeft

Next tactic:
dsimp
theorem hom_left_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actLeft ≫ hom P Q L =
      (𝟙 R.X ⊗ hom P Q L) ≫ (P.tensorBimod (Q.tensorBimod L)).actLeft := 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
⊢ TensorBimod.actLeft (tensorBimod P Q) L ≫ hom P Q L = (𝟙 R.X ⊗ hom P Q L) ≫ TensorBimod.actLeft P (tensorBimod Q L)

Next tactic:
dsimp [hom, homAux]
theorem hom_left_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actLeft ≫ hom P Q L =
      (𝟙 R.X ⊗ hom P Q L) ≫ (P.tensorBimod (Q.tensorBimod L)).actLeft := by
  dsimp; dsimp [hom, homAux]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 2 => 

Proof state:
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 R.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))) ≫
    TensorBimod.actLeft (tensorBimod P 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
| coequalizer.desc
    ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ⊗ 𝟙 Q.X)
          ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))).inv ≫
      coequalizer.desc
        ((α_ P.X Q.X L.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))) ≫
            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)))
        (_ :
          (tensorRight L.X).map (P.actRight ⊗ 𝟙 Q.X) ≫
              (α_ P.X Q.X L.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))) ≫
                  coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                    ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft)) =
            (tensorRight L.X).map ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫
              (α_ P.X Q.X L.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))) ≫
                  coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                    ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft))))
    (_ :
      ((tensorBimod P Q).actRight ⊗ 𝟙 L.X) ≫ homAux P Q L =
        ((α_ (tensorBimod P Q).X T.X L.X).hom ≫ (𝟙 (tensorBimod P Q).X ⊗ L.actLeft)) ≫ homAux P Q L)

Next tactic:
rw [TensorBimod.id_tensor_π_actLeft]
theorem hom_left_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actLeft ≫ hom P Q L =
      (𝟙 R.X ⊗ hom P Q L) ≫ (P.tensorBimod (Q.tensorBimod L)).actLeft := by
  dsimp; dsimp [hom, homAux]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 2 => rw [TensorBimod.id_tensor_π_actLeft]
  slice_lhs 3 4 => 

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
| 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)) ≫
    coequalizer.desc
      ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ⊗ 𝟙 Q.X)
            ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))).inv ≫
        coequalizer.desc
          ((α_ P.X Q.X L.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))) ≫
              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)))
          (_ :
            (tensorRight L.X).map (P.actRight ⊗ 𝟙 Q.X) ≫
                (α_ P.X Q.X L.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))) ≫
                    coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                      ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft)) =
              (tensorRight L.X).map ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫
                (α_ P.X Q.X L.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))) ≫
                    coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                      ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft))))
      (_ :
        ((tensorBimod P Q).actRight ⊗ 𝟙 L.X) ≫ homAux P Q L =
          ((α_ (tensorBimod P Q).X T.X L.X).hom ≫ (𝟙 (tensorBimod P Q).X ⊗ L.actLeft)) ≫ homAux P 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_ R.X (tensorBimod P 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
| (tensorBimod P Q).actLeft ⊗ 𝟙 L.X

Next tactic:
rw [coequalizer.π_desc]
theorem hom_left_act_hom' :
    ((P.tensorBimod Q).tensorBimod L).actLeft ≫ hom P Q L =
      (𝟙 R.X ⊗ hom P Q L) ≫ (P.tensorBimod (Q.tensorBimod L)).actLeft := by
  dsimp; dsimp [hom, homAux]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 2 => rw [TensorBimod.id_tensor_π_actLeft]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_rhs 1 2 => 

Proof state:
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 R.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))) ≫
    (𝟙 R.X ⊗
      coequalizer.desc
        ((PreservesCoequalizer.iso (tensorRight L.X) (P.actRight ⊗ 𝟙 Q.X)
              ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft))).inv ≫
          coequalizer.desc
            ((α_ P.X Q.X L.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))) ≫
                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)))
            (_ :
              (tensorRight L.X).map (P.actRight ⊗ 𝟙 Q.X) ≫
                  (α_ P.X Q.X L.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))) ≫
                      coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                        ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft)) =
                (tensorRight L.X).map ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ≫
                  (α_ P.X Q.X L.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))) ≫
                      coequalizer.π (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
                        ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft))))
        (_ :
          ((tensorBimod P Q).actRight ⊗ 𝟙 L.X) ≫ homAux P Q L =
            ((α_ (tensorBimod P Q).X T.X L.X).hom ≫ (𝟙 (tensorBimod P Q).X ⊗ L.actLeft)) ≫ homAux P 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
| TensorBimod.actLeft P (tensorBimod Q L)

Next tactic:
rw [← id_tensor_comp, coequalizer.π_desc, id_tensor_comp]