l3lab/ntp-mathlib huggingface dataset:
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => 

Proof state:
case a.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| (X.one ⊗ 𝟙 (X.X ⊗ N.X)) ≫ (α_ X.X X.X N.X).inv
case a.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| X.mul ⊗ 𝟙 N.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| 𝟙 X.X ⊗ f.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| (λ_ (X.X ⊗ N.X)).inv

Next tactic:
rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  

Proof state:
case h.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
⊢ (𝟙 X.X ⊗ f.hom) ≫
      colimit.ι (parallelPair (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))) WalkingParallelPair.one =
    (𝟙 X.X ⊗ f.hom) ≫
      (λ_ (X.X ⊗ N.X)).inv ≫
        (((α_ (𝟙_ C) X.X N.X).inv ≫ ((X.one ⊗ 𝟙 X.X) ⊗ 𝟙 N.X)) ≫ (X.mul ⊗ 𝟙 N.X)) ≫
          coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))

Next tactic:
slice_rhs 4 5 => rw [← comp_tensor_id, Mon_.one_mul]
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_rhs 4 5 => 

Proof state:
case a.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| ((X.one ⊗ 𝟙 X.X) ⊗ 𝟙 N.X) ≫ (X.mul ⊗ 𝟙 N.X)
case a.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| 𝟙 X.X ⊗ f.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| (λ_ (X.X ⊗ N.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| (α_ (𝟙_ C) X.X N.X).inv

Next tactic:
rw [← comp_tensor_id, Mon_.one_mul]
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_rhs 4 5 => rw [← comp_tensor_id, Mon_.one_mul]
  

Proof state:
case h.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
⊢ (𝟙 X.X ⊗ f.hom) ≫
      colimit.ι (parallelPair (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))) WalkingParallelPair.one =
    (𝟙 X.X ⊗ f.hom) ≫
      (λ_ (X.X ⊗ N.X)).inv ≫
        (α_ (𝟙_ C) X.X N.X).inv ≫
          ((λ_ X.X).hom ⊗ 𝟙 N.X) ≫ coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))

Next tactic:
have : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 _ := by
    pure_coherence
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_rhs 4 5 => rw [← comp_tensor_id, Mon_.one_mul]
  have : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 _ := 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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
⊢ (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 (X.X ⊗ N.X)

Next tactic:
pure_coherence
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_rhs 4 5 => rw [← comp_tensor_id, Mon_.one_mul]
  have : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 _ := by
    pure_coherence
  

Proof state:
case h.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
this : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 (X.X ⊗ N.X)
⊢ (𝟙 X.X ⊗ f.hom) ≫
      colimit.ι (parallelPair (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))) WalkingParallelPair.one =
    (𝟙 X.X ⊗ f.hom) ≫
      (λ_ (X.X ⊗ N.X)).inv ≫
        (α_ (𝟙_ C) X.X N.X).inv ≫
          ((λ_ X.X).hom ⊗ 𝟙 N.X) ≫ coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))

Next tactic:
slice_rhs 2 4 => rw [this]
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_rhs 4 5 => rw [← comp_tensor_id, Mon_.one_mul]
  have : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 _ := by
    pure_coherence
  slice_rhs 2 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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
this : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 (X.X ⊗ N.X)
| (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
this : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 (X.X ⊗ N.X)
| coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
this : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 (X.X ⊗ N.X)
| 𝟙 X.X ⊗ f.hom

Next tactic:
rw [this]
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_rhs 4 5 => rw [← comp_tensor_id, Mon_.one_mul]
  have : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 _ := by
    pure_coherence
  slice_rhs 2 4 => rw [this]
  

Proof state:
case h.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
this : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 (X.X ⊗ N.X)
⊢ (𝟙 X.X ⊗ f.hom) ≫
      colimit.ι (parallelPair (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))) WalkingParallelPair.one =
    (𝟙 X.X ⊗ f.hom) ≫ 𝟙 (X.X ⊗ N.X) ≫ coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))

Next tactic:
slice_rhs 1 2 => rw [Category.comp_id]
theorem id_whisker_left_bimod {X Y : Mon_ C} {M N : Bimod X Y} (f : M ⟶ N) :
    tensorHom (𝟙 (regular X)) f = (leftUnitorBimod M).hom ≫ f ≫ (leftUnitorBimod N).inv := by
  dsimp [tensorHom, regular, leftUnitorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [LeftUnitorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [LeftUnitorBimod.inv]
  slice_rhs 1 2 => rw [Hom.left_act_hom]
  slice_rhs 2 3 => rw [leftUnitor_inv_naturality]
  slice_rhs 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
  slice_rhs 4 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
  slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
  slice_rhs 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]
  slice_rhs 4 5 => rw [← comp_tensor_id, Mon_.one_mul]
  have : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 _ := by
    pure_coherence
  slice_rhs 2 4 => rw [this]
  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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
this : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 (X.X ⊗ N.X)
| (𝟙 X.X ⊗ f.hom) ≫ 𝟙 (X.X ⊗ N.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
this : (λ_ (X.X ⊗ N.X)).inv ≫ (α_ (𝟙_ C) X.X N.X).inv ≫ ((λ_ X.X).hom ⊗ 𝟙 N.X) = 𝟙 (X.X ⊗ N.X)
| coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))

Next tactic:
rw [Category.comp_id]
theorem comp_whisker_left_bimod {W X Y Z : Mon_ C} (M : Bimod W X) (N : Bimod X Y)
    {P P' : Bimod Y Z} (f : P ⟶ P') :
    tensorHom (𝟙 (M.tensorBimod N)) f =
      (associatorBimod M N P).hom ≫
        tensorHom (𝟙 M) (tensorHom (𝟙 N) f) ≫ (associatorBimod M N P').inv := 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)
W X Y Z : Mon_ C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
⊢ tensorHom (𝟙 (tensorBimod M N)) f =
    (associatorBimod M N P).hom ≫ tensorHom (𝟙 M) (tensorHom (𝟙 N) f) ≫ (associatorBimod M N P').inv

Next tactic:
dsimp [tensorHom, tensorBimod, associatorBimod]
theorem comp_whisker_left_bimod {W X Y Z : Mon_ C} (M : Bimod W X) (N : Bimod X Y)
    {P P' : Bimod Y Z} (f : P ⟶ P') :
    tensorHom (𝟙 (M.tensorBimod N)) f =
      (associatorBimod M N P).hom ≫
        tensorHom (𝟙 M) (tensorHom (𝟙 N) f) ≫ (associatorBimod M N P').inv := by
  dsimp [tensorHom, tensorBimod, associatorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => 

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)
W X Y Z : Mon_ C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
| coequalizer.π (TensorBimod.actRight M N ⊗ 𝟙 P.X)
      ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ (𝟙 (TensorBimod.X M N) ⊗ P.actLeft)) ≫
    colimMap
      (parallelPairHom (TensorBimod.actRight M N ⊗ 𝟙 P.X)
        ((α_ (TensorBimod.X M N) Y.X P.X).hom ≫ (𝟙 (TensorBimod.X M N) ⊗ P.actLeft)) (TensorBimod.actRight M N ⊗ 𝟙 P'.X)
        ((α_ (TensorBimod.X M N) Y.X P'.X).hom ≫ (𝟙 (TensorBimod.X M N) ⊗ P'.actLeft))
        ((𝟙 (TensorBimod.X M N) ⊗ 𝟙 Y.X) ⊗ f.hom) (𝟙 (TensorBimod.X M N) ⊗ f.hom)
        (_ :
          ((mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N)).actRight ⊗ 𝟙 P.X) ≫
              ((𝟙 (mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N))).hom ⊗ f.hom) =
            (((𝟙 (mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N))).hom ⊗ 𝟙 Y.X) ⊗ f.hom) ≫
              ((mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N)).actRight ⊗ 𝟙 P'.X))
        (_ :
          ((α_ (mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N)).X Y.X P.X).hom ≫
                (𝟙 (mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N)).X ⊗ P.actLeft)) ≫
              ((𝟙 (mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N))).hom ⊗ f.hom) =
            (((𝟙 (mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N))).hom ⊗ 𝟙 Y.X) ⊗ f.hom) ≫
              (α_ (mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N)).X Y.X P'.X).hom ≫
                (𝟙 (mk (TensorBimod.X M N) (TensorBimod.actLeft M N) (TensorBimod.actRight M N)).X ⊗ P'.actLeft)))

Next tactic:
rw [ι_colimMap, parallelPairHom_app_one]
theorem comp_whisker_left_bimod {W X Y Z : Mon_ C} (M : Bimod W X) (N : Bimod X Y)
    {P P' : Bimod Y Z} (f : P ⟶ P') :
    tensorHom (𝟙 (M.tensorBimod N)) f =
      (associatorBimod M N P).hom ≫
        tensorHom (𝟙 M) (tensorHom (𝟙 N) f) ≫ (associatorBimod M N P').inv := by
  dsimp [tensorHom, tensorBimod, associatorBimod]
  ext
  apply coequalizer.hom_ext
  dsimp
  slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
  dsimp [TensorBimod.X, AssociatorBimod.hom]
  slice_rhs 1 2 => rw [coequalizer.π_desc]
  dsimp [AssociatorBimod.homAux, AssociatorBimod.inv]
  refine' (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 _
  rw [tensorRight_map]
  slice_rhs 1 3 => rw [π_tensor_id_preserves_coequalizer_inv_desc]
  slice_rhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
  slice_rhs 2 3 => rw [← id_tensor_comp, ι_colimMap, parallelPairHom_app_one]
  

Proof state:
case h.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)
W X Y Z : Mon_ C
M : Bimod W X
N : Bimod X Y
P P' : Bimod Y Z
f : P ⟶ P'
⊢ (coequalizer.π (M.actRight ⊗ 𝟙 N.X) ((α_ M.X X.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft)) ⊗ 𝟙 P.X) ≫
      (𝟙 (coequalizer (M.actRight ⊗ 𝟙 N.X) ((α_ M.X X.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft))) ⊗ f.hom) ≫
        colimit.ι
          (parallelPair (TensorBimod.actRight M N ⊗ 𝟙 P'.X)
            ((α_ (coequalizer (M.actRight ⊗ 𝟙 N.X) ((α_ M.X X.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft))) Y.X P'.X).hom ≫
              (𝟙 (coequalizer (M.actRight ⊗ 𝟙 N.X) ((α_ M.X X.X N.X).hom ≫ (𝟙 M.X ⊗ N.actLeft))) ⊗ P'.actLeft)))
          WalkingParallelPair.one =
    (α_ M.X N.X P.X).hom ≫
      ((𝟙 M.X ⊗
            (𝟙 N.X ⊗ f.hom) ≫
              colimit.ι (parallelPair (N.actRight ⊗ 𝟙 P'.X) ((α_ N.X Y.X P'.X).hom ≫ (𝟙 N.X ⊗ P'.actLeft)))
                WalkingParallelPair.one) ≫
          colimit.ι
            (parallelPair
              (M.actRight ⊗ 𝟙 (coequalizer (N.actRight ⊗ 𝟙 P'.X) ((α_ N.X Y.X P'.X).hom ≫ (𝟙 N.X ⊗ P'.actLeft))))
              ((α_ M.X X.X (coequalizer (N.actRight ⊗ 𝟙 P'.X) ((α_ N.X Y.X P'.X).hom ≫ (𝟙 N.X ⊗ P'.actLeft)))).hom ≫
                (𝟙 M.X ⊗ TensorBimod.actLeft N P')))
            WalkingParallelPair.one) ≫
        coequalizer.desc (AssociatorBimod.invAux M N P')
          (_ :
            (M.actRight ⊗ 𝟙 (tensorBimod N P').X) ≫ AssociatorBimod.invAux M N P' =
              ((α_ M.X X.X (tensorBimod N P').X).hom ≫ (𝟙 M.X ⊗ (tensorBimod N P').actLeft)) ≫
                AssociatorBimod.invAux M N P')

Next tactic:
slice_rhs 3 4 => rw [coequalizer.π_desc]