l3lab/ntp-mathlib huggingface dataset:
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| 𝟙 M₂.X ⊗ N₂.actLeft

Next tactic:
rw [associator_naturality]
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [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)
X Y Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
⊢ (α_ M₁.X Y.X N₁.X).hom ≫ (f.hom ⊗ (𝟙 Y.X ⊗ g.hom) ≫ N₂.actLeft) =
    ((α_ M₁.X Y.X N₁.X).hom ≫ (f.hom ⊗ 𝟙 Y.X ⊗ g.hom)) ≫ (𝟙 M₂.X ⊗ N₂.actLeft)

Next tactic:
slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          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)
X Y Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| (f.hom ⊗ 𝟙 Y.X ⊗ g.hom) ≫ (𝟙 M₂.X ⊗ N₂.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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| (α_ M₁.X Y.X N₁.X).hom

Next tactic:
rw [← tensor_comp, Category.comp_id]
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := 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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
⊢ (tensorBimod M₁ N₁).actLeft ≫
      colimMap
        (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) (M₂.actRight ⊗ 𝟙 N₂.X)
          ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
          (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
          (_ :
            ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
              ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft))) =
    (𝟙 X.X ⊗
        colimMap
          (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))
            (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom)
            (f.hom ⊗ g.hom)
            (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
            (_ :
              ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
                ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)))) ≫
      (tensorBimod M₂ N₂).actLeft

Next tactic:
refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := by
    refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
    

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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
⊢ (tensorLeft X.X).map (coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
      (tensorBimod M₁ N₁).actLeft ≫
        colimMap
          (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))
            (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom)
            (f.hom ⊗ g.hom)
            (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
            (_ :
              ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
                ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft))) =
    (tensorLeft X.X).map (coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
      (𝟙 X.X ⊗
          colimMap
            (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))
              (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom)
              (f.hom ⊗ g.hom)
              (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
              (_ :
                ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
                  ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)))) ≫
        (tensorBimod M₂ N₂).actLeft

Next tactic:
dsimp
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := by
    refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
    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)
X Y Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
⊢ (𝟙 X.X ⊗ coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
      TensorBimod.actLeft M₁ N₁ ≫
        colimMap
          (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))
            (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom)
            (f.hom ⊗ g.hom)
            (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
            (_ :
              ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
                ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft))) =
    (𝟙 X.X ⊗ coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
      (𝟙 X.X ⊗
          colimMap
            (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))
              (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom)
              (f.hom ⊗ g.hom)
              (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
              (_ :
                ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
                  ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)))) ≫
        TensorBimod.actLeft M₂ N₂

Next tactic:
slice_lhs 1 2 => rw [TensorBimod.id_tensor_π_actLeft]
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := by
    refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
    dsimp
    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)
X Y Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| (𝟙 X.X ⊗ coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
    TensorBimod.actLeft M₁ N₁
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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| colimMap
    (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) (M₂.actRight ⊗ 𝟙 N₂.X)
      ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
      (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
      (_ :
        ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
          ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)))

Next tactic:
rw [TensorBimod.id_tensor_π_actLeft]
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := by
    refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
    dsimp
    slice_lhs 1 2 => rw [TensorBimod.id_tensor_π_actLeft]
    

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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
⊢ ((α_ X.X M₁.X N₁.X).inv ≫
        (M₁.actLeft ⊗ 𝟙 N₁.X) ≫ coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
      colimMap
        (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) (M₂.actRight ⊗ 𝟙 N₂.X)
          ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
          (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
          (_ :
            ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
              ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft))) =
    (𝟙 X.X ⊗ coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
      (𝟙 X.X ⊗
          colimMap
            (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))
              (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom)
              (f.hom ⊗ g.hom)
              (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
              (_ :
                ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
                  ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)))) ≫
        TensorBimod.actLeft M₂ N₂

Next tactic:
slice_lhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := by
    refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
    dsimp
    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)
X Y Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫
    colimMap
      (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) (M₂.actRight ⊗ 𝟙 N₂.X)
        ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
        (_ :
          ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
            ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| (α_ X.X M₁.X N₁.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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| M₁.actLeft ⊗ 𝟙 N₁.X

Next tactic:
rw [ι_colimMap, parallelPairHom_app_one]
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := by
    refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
    dsimp
    slice_lhs 1 2 => rw [TensorBimod.id_tensor_π_actLeft]
    slice_lhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
    

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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
⊢ (α_ X.X M₁.X N₁.X).inv ≫
      (M₁.actLeft ⊗ 𝟙 N₁.X) ≫
        (f.hom ⊗ g.hom) ≫
          colimit.ι (parallelPair (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)))
            WalkingParallelPair.one =
    (𝟙 X.X ⊗ coequalizer.π (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))) ≫
      (𝟙 X.X ⊗
          colimMap
            (parallelPairHom (M₁.actRight ⊗ 𝟙 N₁.X) ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft))
              (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)) ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom)
              (f.hom ⊗ g.hom)
              (_ : (M₁.actRight ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.hom) = ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (M₂.actRight ⊗ 𝟙 N₂.X))
              (_ :
                ((α_ M₁.X Y.X N₁.X).hom ≫ (𝟙 M₁.X ⊗ N₁.actLeft)) ≫ (f.hom ⊗ g.hom) =
                  ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) ≫ (α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)))) ≫
        TensorBimod.actLeft M₂ N₂

Next tactic:
slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
/-- Tensor product of two morphisms of bimodule objects. -/
@[simps]
noncomputable def tensorHom {X Y Z : Mon_ C} {M₁ M₂ : Bimod X Y} {N₁ N₂ : Bimod Y Z} (f : M₁ ⟶ M₂)
    (g : N₁ ⟶ N₂) : M₁.tensorBimod N₁ ⟶ M₂.tensorBimod N₂ where
  hom :=
    colimMap
      (parallelPairHom _ _ _ _ ((f.hom ⊗ 𝟙 Y.X) ⊗ g.hom) (f.hom ⊗ g.hom)
        (by
          rw [← tensor_comp, ← tensor_comp, Hom.right_act_hom, Category.id_comp, Category.comp_id])
        (by
          slice_lhs 2 3 => rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]
          slice_rhs 1 2 => rw [associator_naturality]
          slice_rhs 2 3 => rw [← tensor_comp, Category.comp_id]))
  left_act_hom := by
    refine' (cancel_epi ((tensorLeft _).map (coequalizer.π _ _))).1 _
    dsimp
    slice_lhs 1 2 => rw [TensorBimod.id_tensor_π_actLeft]
    slice_lhs 3 4 => rw [ι_colimMap, parallelPairHom_app_one]
    slice_lhs 2 3 => 

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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| (M₁.actLeft ⊗ 𝟙 N₁.X) ≫ (f.hom ⊗ g.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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| colimit.ι (parallelPair (M₂.actRight ⊗ 𝟙 N₂.X) ((α_ M₂.X Y.X N₂.X).hom ≫ (𝟙 M₂.X ⊗ N₂.actLeft)))
    WalkingParallelPair.one
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 Z : Mon_ C
M₁ M₂ : Bimod X Y
N₁ N₂ : Bimod Y Z
f : M₁ ⟶ M₂
g : N₁ ⟶ N₂
| (α_ X.X M₁.X N₁.X).inv

Next tactic:
rw [← tensor_comp, Hom.left_act_hom, Category.id_comp]