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]
  

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 =
    coequalizer.π (X.mul ⊗ 𝟙 M.X) ((α_ X.X X.X M.X).hom ≫ (𝟙 X.X ⊗ M.actLeft)) ≫
      coequalizer.desc M.actLeft
          (_ :
            ((regular X).actRight ⊗ 𝟙 M.X) ≫ M.actLeft =
              ((α_ (regular X).X X.X M.X).hom ≫ (𝟙 (regular X).X ⊗ M.actLeft)) ≫ M.actLeft) ≫
        f.hom ≫ LeftUnitorBimod.inv N

Next tactic:
slice_rhs 1 2 => rw [coequalizer.π_desc]
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 => 

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
| coequalizer.π (X.mul ⊗ 𝟙 M.X) ((α_ X.X X.X M.X).hom ≫ (𝟙 X.X ⊗ M.actLeft)) ≫
    coequalizer.desc M.actLeft
      (_ :
        ((regular X).actRight ⊗ 𝟙 M.X) ≫ M.actLeft =
          ((α_ (regular X).X X.X M.X).hom ≫ (𝟙 (regular X).X ⊗ M.actLeft)) ≫ M.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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| f.hom
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
| LeftUnitorBimod.inv N

Next tactic:
rw [coequalizer.π_desc]
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]
  

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 =
    (M.actLeft ≫ f.hom) ≫ LeftUnitorBimod.inv N

Next tactic:
dsimp [LeftUnitorBimod.inv]
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]
  

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 =
    (M.actLeft ≫ f.hom) ≫
      (λ_ N.X).inv ≫ (X.one ⊗ 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 1 2 => rw [Hom.left_act_hom]
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 => 

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

Next tactic:
rw [Hom.left_act_hom]
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]
  

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) ≫ N.actLeft) ≫ (λ_ N.X).inv) ≫ (X.one ⊗ 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 3 => rw [leftUnitor_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 => 

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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| N.actLeft ≫ (λ_ 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
| X.one ⊗ 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
| 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

Next tactic:
rw [leftUnitor_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]
  

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 ≫ (𝟙 (𝟙_ C) ⊗ N.actLeft)) ≫ (X.one ⊗ 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 3 4 => rw [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
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 => 

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)
X Y : Mon_ C
M N : Bimod X Y
f : M ⟶ N
| (𝟙 (𝟙_ C) ⊗ N.actLeft) ≫ (X.one ⊗ 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 [id_tensor_comp_tensor_id, ← tensor_id_comp_id_tensor]
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]
  

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 ≫
        ((X.one ⊗ 𝟙 (X.X ⊗ N.X)) ≫ (𝟙 X.X ⊗ N.actLeft)) ≫
          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 4 => rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
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 => 

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
| 𝟙 X.X ⊗ N.actLeft
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
| X.one ⊗ 𝟙 (X.X ⊗ N.X)

Next tactic:
rw [← Iso.inv_hom_id_assoc (α_ X.X X.X N.X) (𝟙 X.X ⊗ N.actLeft)]
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)]
  

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 ≫
        (X.one ⊗ 𝟙 (X.X ⊗ N.X)) ≫
          ((α_ X.X X.X N.X).inv ≫ (α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft)) ≫
            coequalizer.π (X.mul ⊗ 𝟙 N.X) ((α_ X.X X.X N.X).hom ≫ (𝟙 X.X ⊗ N.actLeft))

Next tactic:
slice_rhs 5 7 => rw [← Category.assoc, ← coequalizer.condition]
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 => 

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
| (α_ X.X X.X N.X).hom ≫
    (𝟙 X.X ⊗ N.actLeft) ≫ 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
| X.one ⊗ 𝟙 (X.X ⊗ 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
| (α_ X.X X.X N.X).inv

Next tactic:
rw [← Category.assoc, ← coequalizer.condition]
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]
  

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 ≫
        (X.one ⊗ 𝟙 (X.X ⊗ N.X)) ≫
          (α_ X.X X.X N.X).inv ≫
            (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 3 4 => rw [← MonoidalCategory.tensor_id, associator_inv_naturality]