l3lab/ntp-mathlib huggingface dataset:
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ := by
  dsimp [hom, homAux, inv, invAux]
  apply coequalizer.hom_ext
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 3 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  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.actRight P Q ⊗ 𝟙 L.X)
      ((α_ (TensorBimod.X P Q) T.X L.X).hom ≫ (𝟙 (TensorBimod.X P Q) ⊗ 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_ 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

Next tactic:
rw [coequalizer.π_desc]
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ := by
  dsimp [hom, homAux, inv, invAux]
  apply coequalizer.hom_ext
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 3 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  

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

Next tactic:
slice_lhs 2 4 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ := by
  dsimp [hom, homAux, inv, invAux]
  apply coequalizer.hom_ext
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 3 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_lhs 2 4 => 

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
| (coequalizer.π (P.actRight ⊗ 𝟙 Q.X) ((α_ P.X S.X Q.X).hom ≫ (𝟙 P.X ⊗ Q.actLeft)) ⊗ 𝟙 L.X) ≫
    (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)))
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).inv

Next tactic:
rw [π_tensor_id_preserves_coequalizer_inv_desc]
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ := by
  dsimp [hom, homAux, inv, invAux]
  apply coequalizer.hom_ext
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 3 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_lhs 2 4 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  

Proof state:
case h
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).inv ≫
      (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)) =
    (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)) ≫
        𝟙 (TensorBimod.X P (tensorBimod Q L))

Next tactic:
slice_lhs 1 3 => rw [Iso.inv_hom_id_assoc]
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ := by
  dsimp [hom, homAux, inv, invAux]
  apply coequalizer.hom_ext
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 3 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_lhs 2 4 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_lhs 1 3 => 

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
| (α_ P.X Q.X L.X).inv ≫
    (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)))
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 (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:
rw [Iso.inv_hom_id_assoc]
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ := by
  dsimp [hom, homAux, inv, invAux]
  apply coequalizer.hom_ext
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 3 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_lhs 2 4 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_lhs 1 3 => rw [Iso.inv_hom_id_assoc]
  

Proof state:
case h
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))) ≫
      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)) =
    (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)) ≫
        𝟙 (TensorBimod.X P (tensorBimod Q L))

Next tactic:
dsimp only [TensorBimod.X]
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ := by
  dsimp [hom, homAux, inv, invAux]
  apply coequalizer.hom_ext
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 3 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_lhs 2 4 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_lhs 1 3 => rw [Iso.inv_hom_id_assoc]
  dsimp only [TensorBimod.X]
  

Proof state:
case h
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))) ≫
      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)) =
    (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)) ≫
        𝟙
          (coequalizer (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
            ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q L).actLeft)))

Next tactic:
slice_rhs 2 3 => rw [Category.comp_id]
theorem inv_hom_id : inv P Q L ≫ hom P Q L = 𝟙 _ := by
  dsimp [hom, homAux, inv, invAux]
  apply coequalizer.hom_ext
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
  rw [tensorLeft_map]
  slice_lhs 1 3 => rw [id_tensor_π_preserves_coequalizer_inv_desc]
  slice_lhs 3 4 => rw [coequalizer.π_desc]
  slice_lhs 2 4 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_lhs 1 3 => rw [Iso.inv_hom_id_assoc]
  dsimp only [TensorBimod.X]
  slice_rhs 2 3 => 

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
| 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)) ≫
    𝟙
      (coequalizer (P.actRight ⊗ 𝟙 (tensorBimod Q L).X)
        ((α_ P.X S.X (tensorBimod Q L).X).hom ≫ (𝟙 P.X ⊗ (tensorBimod Q 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 ⊗ coequalizer.π (Q.actRight ⊗ 𝟙 L.X) ((α_ Q.X T.X L.X).hom ≫ (𝟙 Q.X ⊗ L.actLeft))

Next tactic:
rw [Category.comp_id]
/-- The underlying morphism of the forward component of the left unitor isomorphism. -/
noncomputable def hom : TensorBimod.X (regular R) P ⟶ P.X :=
  coequalizer.desc P.actLeft (by 

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
⊢ ((regular R).actRight ⊗ 𝟙 P.X) ≫ P.actLeft =
    ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) ≫ P.actLeft

Next tactic:
dsimp
/-- The underlying morphism of the forward component of the left unitor isomorphism. -/
noncomputable def hom : TensorBimod.X (regular R) P ⟶ P.X :=
  coequalizer.desc P.actLeft (by dsimp; 

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
⊢ (R.mul ⊗ 𝟙 P.X) ≫ P.actLeft = ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) ≫ P.actLeft

Next tactic:
rw [Category.assoc, left_assoc]
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
⊢ hom P ≫ inv P = 𝟙 (TensorBimod.X (regular R) P)

Next tactic:
dsimp only [hom, inv, TensorBimod.X]
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  dsimp only [hom, inv, TensorBimod.X]
  

Proof state:
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
⊢ coequalizer.desc P.actLeft
        (_ :
          ((regular R).actRight ⊗ 𝟙 P.X) ≫ P.actLeft =
            ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) ≫ P.actLeft) ≫
      (λ_ P.X).inv ≫
        (R.one ⊗ 𝟙 P.X) ≫
          coequalizer.π ((regular R).actRight ⊗ 𝟙 P.X)
            ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) =
    𝟙 (coequalizer ((regular R).actRight ⊗ 𝟙 P.X) ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)))

Next tactic:
ext
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  dsimp only [hom, inv, TensorBimod.X]
  ext; 

Proof state:
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
⊢ coequalizer.π ((regular R).actRight ⊗ 𝟙 P.X) ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) ≫
      coequalizer.desc P.actLeft
          (_ :
            ((regular R).actRight ⊗ 𝟙 P.X) ≫ P.actLeft =
              ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) ≫ P.actLeft) ≫
        (λ_ P.X).inv ≫
          (R.one ⊗ 𝟙 P.X) ≫
            coequalizer.π ((regular R).actRight ⊗ 𝟙 P.X)
              ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) =
    coequalizer.π ((regular R).actRight ⊗ 𝟙 P.X) ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) ≫
      𝟙 (coequalizer ((regular R).actRight ⊗ 𝟙 P.X) ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)))

Next tactic:
dsimp
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  dsimp only [hom, inv, TensorBimod.X]
  ext; dsimp
  

Proof state:
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
⊢ coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) ≫
      coequalizer.desc P.actLeft
          (_ :
            ((regular R).actRight ⊗ 𝟙 P.X) ≫ P.actLeft =
              ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) ≫ P.actLeft) ≫
        (λ_ P.X).inv ≫ (R.one ⊗ 𝟙 P.X) ≫ coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) =
    coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) ≫
      𝟙 (coequalizer (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)))

Next tactic:
slice_lhs 1 2 => rw [coequalizer.π_desc]
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  dsimp only [hom, inv, TensorBimod.X]
  ext; dsimp
  slice_lhs 1 2 => 

Proof state:
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) ≫
    coequalizer.desc P.actLeft
      (_ :
        ((regular R).actRight ⊗ 𝟙 P.X) ≫ P.actLeft =
          ((α_ (regular R).X R.X P.X).hom ≫ (𝟙 (regular R).X ⊗ P.actLeft)) ≫ P.actLeft)
case a.a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| (λ_ P.X).inv
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| R.one ⊗ 𝟙 P.X
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft))

Next tactic:
rw [coequalizer.π_desc]
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  dsimp only [hom, inv, TensorBimod.X]
  ext; dsimp
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  

Proof state:
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
⊢ ((P.actLeft ≫ (λ_ P.X).inv) ≫ (R.one ⊗ 𝟙 P.X)) ≫
      coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) =
    coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) ≫
      𝟙 (coequalizer (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)))

Next tactic:
slice_lhs 1 2 => rw [leftUnitor_inv_naturality]
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  dsimp only [hom, inv, TensorBimod.X]
  ext; dsimp
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  slice_lhs 1 2 => 

Proof state:
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| P.actLeft ≫ (λ_ P.X).inv
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| R.one ⊗ 𝟙 P.X
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft))

Next tactic:
rw [leftUnitor_inv_naturality]
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  dsimp only [hom, inv, TensorBimod.X]
  ext; dsimp
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  slice_lhs 1 2 => rw [leftUnitor_inv_naturality]
  

Proof state:
case h
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
⊢ (((λ_ (R.X ⊗ P.X)).inv ≫ (𝟙 (𝟙_ C) ⊗ P.actLeft)) ≫ (R.one ⊗ 𝟙 P.X)) ≫
      coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) =
    coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)) ≫
      𝟙 (coequalizer (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft)))

Next tactic:
slice_lhs 2 3 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
theorem hom_inv_id : hom P ≫ inv P = 𝟙 _ := by
  dsimp only [hom, inv, TensorBimod.X]
  ext; dsimp
  slice_lhs 1 2 => rw [coequalizer.π_desc]
  slice_lhs 1 2 => rw [leftUnitor_inv_naturality]
  slice_lhs 2 3 => 

Proof state:
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| (𝟙 (𝟙_ C) ⊗ P.actLeft) ≫ (R.one ⊗ 𝟙 P.X)
case a.a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| coequalizer.π (R.mul ⊗ 𝟙 P.X) ((α_ R.X R.X P.X).hom ≫ (𝟙 R.X ⊗ P.actLeft))
case a
C : Type u₁
inst✝² : Category.{v₁, u₁} C
inst✝¹ : MonoidalCategory C
A B : Mon_ C
M : Bimod A B
inst✝ : HasCoequalizers C
R S : Mon_ C
P : Bimod R S
| (λ_ (R.X ⊗ P.X)).inv

Next tactic:
rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]