awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    ((((f.hom ▷ Y.X ≫ N.actRight) ≫ Iso.inv (ρ_ N.X)) ≫ N.X ◁ η) ≫
      coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    ((M.actRight ≫ f.hom) ≫
      Iso.inv (ρ_ N.X) ≫ N.X ◁ η ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))

Tactic:
slice_rhs 1 2 => rw [Hom.right_act_hom]

Premises used: Bimod.Hom.right_act_hom
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫ N.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst)
| N.actRight ≫ Iso.inv (ρ_ N.X)

Tactic:
rw [rightUnitor_inv_naturality]

Premises used: CategoryTheory.MonoidalCategory.rightUnitor_inv_naturality
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      ((Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫ N.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫
          N.X ◁ η) ≫
        coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    ((((f.hom ▷ Y.X ≫ N.actRight) ≫ Iso.inv (ρ_ N.X)) ≫ N.X ◁ η) ≫
      coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))

Tactic:
slice_rhs 2 3 => rw [rightUnitor_inv_naturality]

Premises used: CategoryTheory.MonoidalCategory.rightUnitor_inv_naturality
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫ N.actRight ▷ Y.X
| N.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫ N.X ◁ η

Tactic:
rw [← whisker_exchange]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫ N.actRight ▷ Y.X) ≫
          coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      ((Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫ N.actRight ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst)) ≫
          N.X ◁ η) ≫
        coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))

Tactic:
slice_rhs 3 4 => rw [← whisker_exchange]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)
| N.actRight ▷ Y.X ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)

Tactic:
rw [coequalizer.condition]

Premises used: CategoryTheory.Limits.coequalizer.condition
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫
          (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫ N.actRight ▷ Y.X) ≫
          coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))

Tactic:
slice_rhs 4 5 => rw [coequalizer.condition]

Premises used: CategoryTheory.Limits.coequalizer.condition
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ N.X ◁ Y.X ◁ η
| tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫ Iso.hom (α_ N.X Y.X Y.X)

Tactic:
rw [associator_naturality_right]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_right
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        ((Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ N.X ◁ Y.X ◁ η) ≫ N.X ◁ μ) ≫
          coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X ◁ η ≫
          (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))

Tactic:
slice_rhs 3 4 => rw [associator_naturality_right]

Premises used: CategoryTheory.MonoidalCategory.associator_naturality_right
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
a : N.X ◁ Iso.hom (ρ_ Y.X)
| N.X ◁ Y.X ◁ η ≫ N.X ◁ μ

Tactic:
rw [← whiskerLeft_comp, MonObj.mul_one]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonObj.mul_one
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
h :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫
          N.X ◁ Iso.hom (ρ_ Y.X) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        ((Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫ N.X ◁ Y.X ◁ η) ≫ N.X ◁ μ) ≫
          coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))

Tactic:
slice_rhs 4 5 => rw [← whiskerLeft_comp, MonObj.mul_one]

Premises used: CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonObj.mul_one
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
X Y : Mon.{v₁, u₁} C
M N : Bimod X Y
f : M ⟶ N
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) M.X Y.X ⟶
      TensorBimod.X N
        { X := Y.X, actLeft := μ, one_actLeft := ⋯, left_assoc := ⋯, actRight := μ, actRight_one := ⋯, right_assoc := ⋯,
          middle_assoc := ⋯ : Bimod Y Y })
    (f.hom ▷ Y.X ≫
      colimit.ι (parallelPair (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ)) WalkingParallelPair.one)
    (f.hom ▷ Y.X ≫
      Iso.inv (ρ_ (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X)) ≫
        Iso.hom (α_ N.X Y.X (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ≫
          N.X ◁ Iso.hom (ρ_ Y.X) ≫ coequalizer.π (N.actRight ▷ Y.X) (Iso.hom (α_ N.X Y.X Y.X) ≫ N.X ◁ μ))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
dsimp :
  whiskerRight f
      { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
        actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ } =
    Iso.inv
        (isoOfIso
          { hom := AssociatorBimod.hom M N P, inv := AssociatorBimod.inv M N P, hom_inv_id := ⋯, inv_hom_id := ⋯ :
            TensorBimod.X
                { X := TensorBimod.X M N, actLeft := TensorBimod.actLeft M N, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight M N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                  Bimod W Y }
                P ≅
              TensorBimod.X M
                { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
                  actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                  Bimod X Z } }
          ⋯ ⋯) ≫
      whiskerRight (whiskerRight f N) P ≫
        Iso.hom
          (isoOfIso
            { hom := AssociatorBimod.hom M' N P, inv := AssociatorBimod.inv M' N P, hom_inv_id := ⋯, inv_hom_id := ⋯ :
              TensorBimod.X
                  { X := TensorBimod.X M' N, actLeft := TensorBimod.actLeft M' N, one_actLeft := ⋯, left_assoc := ⋯,
                    actRight := TensorBimod.actRight M' N, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                    Bimod W Y }
                  P ≅
                TensorBimod.X M'
                  { X := TensorBimod.X N P, actLeft := TensorBimod.actLeft N P, one_actLeft := ⋯, left_assoc := ⋯,
                    actRight := TensorBimod.actRight N P, actRight_one := ⋯, right_assoc := ⋯, middle_assoc := ⋯ :
                    Bimod X Z } }
            ⋯ ⋯)
⊢ whiskerRight f (N.tensorBimod P) =
    Iso.inv (M.associatorBimod N P) ≫ whiskerRight (whiskerRight f N) P ≫ Iso.hom (M'.associatorBimod N P)

Tactic:
dsimp [tensorHom, tensorBimod, associatorBimod]

Premises used: Bimod.tensorBimod, Bimod.associatorBimod, CategoryTheory.MonoidalCategoryStruct.tensorHom
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
rw :
  f.hom ▷ TensorBimod.X N P ≫
    colimit.ι
      (parallelPair (X := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M'.X X.X) (TensorBimod.X N P))
        (M'.actRight ▷ TensorBimod.X N P) (Iso.hom (α_ M'.X X.X (TensorBimod.X N P)) ≫ M'.X ◁ TensorBimod.actLeft N P))
      WalkingParallelPair.one
| CategoryStruct.comp (Y :=
    coequalizer (X := tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) M.X X.X) (TensorBimod.X N P))
      (M.actRight ▷ TensorBimod.X N P) (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P))
    (coequalizer.π (M.actRight ▷ TensorBimod.X N P)
      (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P))
    (colimMap
      (parallelPairHom (M.actRight ▷ TensorBimod.X N P)
        (Iso.hom (α_ M.X X.X (TensorBimod.X N P)) ≫ M.X ◁ TensorBimod.actLeft N P) (M'.actRight ▷ TensorBimod.X N P)
        (Iso.hom (α_ M'.X X.X (TensorBimod.X N P)) ≫ M'.X ◁ TensorBimod.actLeft N P) (f.hom ▷ X.X ▷ TensorBimod.X N P)
        (f.hom ▷ TensorBimod.X N P) ⋯ ⋯))

Tactic:
rw [ι_colimMap, parallelPairHom_app_one]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.Limits.parallelPairHom_app_one
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
a :
  Iso.inv (α_ M.X N.X P.X) ≫
    coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X ≫
      coequalizer.π (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft)
| (CategoryStruct.comp (Y :=
      Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft M.X)
        (coequalizer (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))
      (M.X ◁ coequalizer.π (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft))
      (CategoryStruct.comp (Y :=
        coequalizer (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) M.X
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X Y.X) P.X))
          (M.X ◁ N.actRight ▷ P.X) (M.X ◁ (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))
        (Iso.inv
          (PreservesCoequalizer.iso (tensorLeft M.X) (N.actRight ▷ P.X) (Iso.hom (α_ N.X Y.X P.X) ≫ N.X ◁ P.actLeft)))
        (coequalizer.desc
          (Iso.inv (α_ M.X N.X P.X) ≫
            coequalizer.π (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ▷ P.X ≫
              coequalizer.π (TensorBimod.actRight M N ▷ P.X)
                (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
          ⋯)) :
    Functor.obj.{v₁, v₁, u₁, u₁} (tensorLeft M.X) (tensorObj.{v₁, u₁} (𝒞 := inst) N.X P.X) ⟶
      coequalizer (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) Y.X P.X) ≫
          coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ P.actLeft))

Tactic:
rw [id_tensor_π_preserves_coequalizer_inv_desc]

Premises used: id_tensor_π_preserves_coequalizer_inv_desc
-- Module: Mathlib.CategoryTheory.Monoidal.Bimod
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
inst_2 : HasCoequalizers.{v₁, u₁} C
inst_3 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorLeft X)
inst_4 : ∀ (X : C), PreservesColimitsOfSize.{0, 0, v₁, v₁, u₁, u₁} (tensorRight X)
W X Y Z : Mon.{v₁, u₁} C
M M' : Bimod W X
f : M ⟶ M'
N : Bimod X Y
P : Bimod Y Z
a :
  colimMap
        (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
          (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
      P.X ≫
    colimit.ι
      (parallelPair (TensorBimod.actRight M' N ▷ P.X)
        (Iso.hom (α_ (coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft)) Y.X P.X) ≫
          coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) ◁ P.actLeft))
      WalkingParallelPair.one
| (CategoryStruct.comp (Y :=
      coequalizer (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) Y.X P.X) ≫
          coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ P.actLeft))
      (coequalizer.π (TensorBimod.actRight M N ▷ P.X)
        (Iso.hom (α_ (TensorBimod.X M N) Y.X P.X) ≫ TensorBimod.X M N ◁ P.actLeft))
      (colimMap
        (parallelPairHom (TensorBimod.actRight M N ▷ P.X)
          (Iso.hom (α_ (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) Y.X P.X) ≫
            coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) ◁ P.actLeft)
          (TensorBimod.actRight M' N ▷ P.X)
          (Iso.hom (α_ (coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft)) Y.X P.X) ≫
            coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) ◁ P.actLeft)
          (colimMap
                (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
                  (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
              Y.X ▷
            P.X)
          (colimMap
              (parallelPairHom (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft) (M'.actRight ▷ N.X)
                (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) (f.hom ▷ X.X ▷ N.X) (f.hom ▷ N.X) ⋯ ⋯) ▷
            P.X)
          ⋯ ⋯)) :
    tensorObj.{v₁, u₁} (𝒞 := inst) (coequalizer (M.actRight ▷ N.X) (Iso.hom (α_ M.X X.X N.X) ≫ M.X ◁ N.actLeft)) P.X ⟶
      coequalizer (TensorBimod.actRight M' N ▷ P.X)
        (Iso.hom (α_ (coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft)) Y.X P.X) ≫
          coequalizer (M'.actRight ▷ N.X) (Iso.hom (α_ M'.X X.X N.X) ≫ M'.X ◁ N.actLeft) ◁ P.actLeft))

Tactic:
rw [ι_colimMap, parallelPairHom_app_one]

Premises used: CategoryTheory.Limits.ι_colimMap, CategoryTheory.Limits.parallelPairHom_app_one