awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
inst_3 : HasRightDual.{v₁, u₁} Y
f : X ⟶ Y
⊢ Eq (α :=
    @rightDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (Iso.inv (ρ_ (@rightDual.{v₁, u₁} _ inst inst_1 Y _)) ≫
      @rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
        Iso.inv (α_ (@rightDual.{v₁, u₁} _ inst inst_1 Y _) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ≫
          (@rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ f ≫ ε_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)) ▷
            @rightDual.{v₁, u₁} _ inst inst_1 X _)
    (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
        (Iso.inv (ρ_ (@rightDual.{v₁, u₁} _ inst inst_1 Y _)))
        (@rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
          @rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ f ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _ ≫
            Iso.inv (α_ (@rightDual.{v₁, u₁} _ inst inst_1 Y _) Y (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ≫
              ε_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _ ≫
                Iso.hom (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _))) ≫
      Iso.inv (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _)))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
f : Y ⟶ @rightDual.{v₁, u₁} _ inst inst_1 X _
dsimp :
  Eq (α :=
    Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (Iso.inv (ρ_ Y) ≫
      Y ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
        Iso.inv (α_ Y X (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ≫
          (f ▷ X ≫ ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)
    (monoidalComp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) Y (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟙 Y)
      (monoidalComp (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
        (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X)
          (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (Y ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
          f ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)))
⊢ Eq (α :=
    Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    ((tensorRightHomEquiv Y X (@rightDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : Y ⊗ X ⟶ 𝟙_ C) → Y ⟶ 𝟙_ C ⊗ Xᘁ)
        (f ▷ X ≫ ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _)) :
      Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (monoidalComp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) Y (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟙 Y)
        (monoidalComp (X :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _)
            (tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
          (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X)
            (@rightDual.{v₁, u₁} _ inst inst_1 X _))
          (Y ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
            f ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _))
          (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)) :
      Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))

Tactic:
dsimp [tensorRightHomEquiv]

Premises used: CategoryTheory.tensorRightHomEquiv
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
f : Y ⟶ @rightDual.{v₁, u₁} _ inst inst_1 X _
⊢ Eq (α :=
    Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (Iso.inv (ρ_ Y) ≫
      Y ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
        Iso.inv (α_ Y X (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ≫
          (f ▷ X ≫ ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)
    (monoidalComp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) Y (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟙 Y)
      (monoidalComp (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
        (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X)
          (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (Y ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
          f ▷ tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)))

Tactic:
monoidal
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
f : Y ⟶ @rightDual.{v₁, u₁} _ inst inst_1 X _
⊢ Eq (α :=
    Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (monoidalComp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) Y (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟙 Y)
      (monoidalComp (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
        (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X)
          (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (f ▷ tensorUnit.{v₁, u₁} C (𝒞 := inst) ≫
          @rightDual.{v₁, u₁} _ inst inst_1 X _ ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)))
    (monoidalComp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst))) f
      (monoidalComp (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
        (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X)
          (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (@rightDual.{v₁, u₁} _ inst inst_1 X _ ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)))

Tactic:
monoidal
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
f : Y ⟶ @rightDual.{v₁, u₁} _ inst inst_1 X _
rw :
  Eq (α :=
    Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (monoidalComp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst))) f
      (Iso.hom ⊗𝟙 ))
    (f ≫ Iso.inv (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
⊢ Eq (α :=
    Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (monoidalComp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst))) f
      (monoidalComp (X :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
        (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X)
          (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (@rightDual.{v₁, u₁} _ inst inst_1 X _ ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _))
        (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)))
    (f ≫ Iso.inv (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _)))

Tactic:
rw [coevaluation_evaluation'']

Premises used: CategoryTheory.ExactPairing.coevaluation_evaluation''
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
f : Y ⟶ @rightDual.{v₁, u₁} _ inst inst_1 X _
⊢ Eq (α :=
    Y ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (monoidalComp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst))) f
      (Iso.hom ⊗𝟙 ))
    (f ≫ Iso.inv (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _)))

Tactic:
monoidal
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
inst_3 : HasRightDual.{v₁, u₁} Y
f : X ⟶ Y
apply_fun :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
      @rightDual.{v₁, u₁} _ inst inst_1 X _)
    ((Equiv.symm
            (tensorLeftHomEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)
              (@rightDual.{v₁, u₁} _ inst inst_1 X _)) :
          (a : 𝟙_ C ⟶ Y ⊗ Xᘁ) → Yᘁ ⊗ 𝟙_ C ⟶ Xᘁ)
        (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _))
          (η_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)) (Y ◁ fᘁ)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
        @rightDual.{v₁, u₁} _ inst inst_1 X _)
    ((Equiv.symm
            (tensorLeftHomEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)
              (@rightDual.{v₁, u₁} _ inst inst_1 X _)) :
          (a : 𝟙_ C ⟶ Y ⊗ Xᘁ) → Yᘁ ⊗ 𝟙_ C ⟶ Xᘁ)
        (η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫ f ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
        @rightDual.{v₁, u₁} _ inst inst_1 X _)
⊢ Eq (α := tensorUnit.{v₁, u₁} C (𝒞 := inst) ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) Y (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _))
      (η_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)) (Y ◁ fᘁ))
    (η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫ f ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _)

Tactic:
apply_fun (tensorLeftHomEquiv _ Y (Yᘁ) _).symm

Premises used: CategoryTheory.HasRightDual.rightDual, CategoryTheory.tensorLeftHomEquiv, Equiv.symm
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
inst_3 : HasRightDual.{v₁, u₁} Y
f : X ⟶ Y
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
      @rightDual.{v₁, u₁} _ inst inst_1 X _)
    ((Equiv.symm
            (tensorLeftHomEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)
              (@rightDual.{v₁, u₁} _ inst inst_1 X _)) :
          (a : 𝟙_ C ⟶ Y ⊗ Xᘁ) → Yᘁ ⊗ 𝟙_ C ⟶ Xᘁ)
        (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _))
          (η_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)) (Y ◁ fᘁ)) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
        @rightDual.{v₁, u₁} _ inst inst_1 X _)
    ((Equiv.symm
            (tensorLeftHomEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)
              (@rightDual.{v₁, u₁} _ inst inst_1 X _)) :
          (a : 𝟙_ C ⟶ Y ⊗ Xᘁ) → Yᘁ ⊗ 𝟙_ C ⟶ Xᘁ)
        (η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫ f ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
        @rightDual.{v₁, u₁} _ inst inst_1 X _)

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasLeftDual.{v₁, u₁} X
inst_3 : HasLeftDual.{v₁, u₁} Y
f : X ⟶ Y
apply_fun :
  Eq (α :=
    @leftDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    ((tensorLeftHomEquiv (@leftDual.{v₁, u₁} _ inst inst_1 Y _) (@leftDual.{v₁, u₁} _ inst inst_1 X _) X
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : X ⊗ ᘁY ⟶ 𝟙_ C) → ᘁY ⟶ ᘁX ⊗ 𝟙_ C)
        ((X ◁ ᘁf) ≫ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X) :
      @leftDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    ((tensorLeftHomEquiv (@leftDual.{v₁, u₁} _ inst inst_1 Y _) (@leftDual.{v₁, u₁} _ inst inst_1 X _) X
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : X ⊗ ᘁY ⟶ 𝟙_ C) → ᘁY ⟶ ᘁX ⊗ 𝟙_ C)
        (f ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ≫ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y) :
      @leftDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) X (@leftDual.{v₁, u₁} _ inst inst_1 Y _) ⟶ tensorUnit.{v₁, u₁} C (𝒞 := inst))
    ((X ◁ ᘁf) ≫ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X)
    (f ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ≫ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y)

Tactic:
apply_fun tensorLeftHomEquiv _ (ᘁX) X _

Premises used: CategoryTheory.tensorLeftHomEquiv, CategoryTheory.HasLeftDual.leftDual
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasLeftDual.{v₁, u₁} X
inst_3 : HasLeftDual.{v₁, u₁} Y
f : X ⟶ Y
⊢ Eq (α :=
    @leftDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    ((tensorLeftHomEquiv (@leftDual.{v₁, u₁} _ inst inst_1 Y _) (@leftDual.{v₁, u₁} _ inst inst_1 X _) X
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : X ⊗ ᘁY ⟶ 𝟙_ C) → ᘁY ⟶ ᘁX ⊗ 𝟙_ C)
        ((X ◁ ᘁf) ≫ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X) :
      @leftDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    ((tensorLeftHomEquiv (@leftDual.{v₁, u₁} _ inst inst_1 Y _) (@leftDual.{v₁, u₁} _ inst inst_1 X _) X
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : X ⊗ ᘁY ⟶ 𝟙_ C) → ᘁY ⟶ ᘁX ⊗ 𝟙_ C)
        (f ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ≫ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y) :
      @leftDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasLeftDual.{v₁, u₁} X
inst_3 : HasLeftDual.{v₁, u₁} Y
f : X ⟶ Y
apply_fun :
  Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) ⟶
      @leftDual.{v₁, u₁} _ inst inst_1 X _)
    ((Equiv.symm
            (tensorRightHomEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y
              (@leftDual.{v₁, u₁} _ inst inst_1 X _)) :
          (a : 𝟙_ C ⟶ ᘁX ⊗ Y) → 𝟙_ C ⊗ ᘁY ⟶ ᘁX)
        (η_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫ (ᘁf) ▷ Y) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) ⟶
        @leftDual.{v₁, u₁} _ inst inst_1 X _)
    ((Equiv.symm
            (tensorRightHomEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y
              (@leftDual.{v₁, u₁} _ inst inst_1 X _)) :
          (a : 𝟙_ C ⟶ ᘁX ⊗ Y) → 𝟙_ C ⊗ ᘁY ⟶ ᘁX)
        (η_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X ≫ @leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ f) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) ⟶
        @leftDual.{v₁, u₁} _ inst inst_1 X _)
⊢ Eq (α := tensorUnit.{v₁, u₁} C (𝒞 := inst) ⟶ tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _) Y)
    (η_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫ (ᘁf) ▷ Y)
    (η_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X ≫ @leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ f)

Tactic:
apply_fun (tensorRightHomEquiv _ (ᘁY) Y _).symm

Premises used: CategoryTheory.tensorRightHomEquiv, CategoryTheory.HasLeftDual.leftDual, Equiv.symm
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasLeftDual.{v₁, u₁} X
inst_3 : HasLeftDual.{v₁, u₁} Y
f : X ⟶ Y
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) ⟶
      @leftDual.{v₁, u₁} _ inst inst_1 X _)
    ((Equiv.symm
            (tensorRightHomEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y
              (@leftDual.{v₁, u₁} _ inst inst_1 X _)) :
          (a : 𝟙_ C ⟶ ᘁX ⊗ Y) → 𝟙_ C ⊗ ᘁY ⟶ ᘁX)
        (η_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫ (ᘁf) ▷ Y) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) ⟶
        @leftDual.{v₁, u₁} _ inst inst_1 X _)
    ((Equiv.symm
            (tensorRightHomEquiv (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y
              (@leftDual.{v₁, u₁} _ inst inst_1 X _)) :
          (a : 𝟙_ C ⟶ ᘁX ⊗ Y) → 𝟙_ C ⊗ ᘁY ⟶ ᘁX)
        (η_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X ≫ @leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ f) :
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) ⟶
        @leftDual.{v₁, u₁} _ inst inst_1 X _)

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
inst_3 : HasRightDual.{v₁, u₁} Y
f : X ⟶ Y
apply_fun :
  Eq (α :=
    @rightDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    ((tensorRightHomEquiv (@rightDual.{v₁, u₁} _ inst inst_1 Y _) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : Yᘁ ⊗ X ⟶ 𝟙_ C) → Yᘁ ⟶ 𝟙_ C ⊗ Xᘁ)
        (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X) (fᘁ ▷ X)
          (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _))) :
      @rightDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    ((tensorRightHomEquiv (@rightDual.{v₁, u₁} _ inst inst_1 Y _) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : Yᘁ ⊗ X ⟶ 𝟙_ C) → Yᘁ ⟶ 𝟙_ C ⊗ Xᘁ)
        (@rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ f ≫ ε_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)) :
      @rightDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
⊢ Eq (α := tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _) X ⟶ tensorUnit.{v₁, u₁} C (𝒞 := inst))
    (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X) (fᘁ ▷ X)
      (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
    (@rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ f ≫ ε_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _))

Tactic:
apply_fun tensorRightHomEquiv _ X (Xᘁ) _

Premises used: CategoryTheory.HasRightDual.rightDual, CategoryTheory.tensorRightHomEquiv
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : C
inst_2 : HasRightDual.{v₁, u₁} X
inst_3 : HasRightDual.{v₁, u₁} Y
f : X ⟶ Y
⊢ Eq (α :=
    @rightDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    ((tensorRightHomEquiv (@rightDual.{v₁, u₁} _ inst inst_1 Y _) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : Yᘁ ⊗ X ⟶ 𝟙_ C) → Yᘁ ⟶ 𝟙_ C ⊗ Xᘁ)
        (CategoryStruct.comp (Y := tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 X _) X) (fᘁ ▷ X)
          (ε_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _))) :
      @rightDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))
    ((tensorRightHomEquiv (@rightDual.{v₁, u₁} _ inst inst_1 Y _) X (@rightDual.{v₁, u₁} _ inst inst_1 X _)
            (tensorUnit.{v₁, u₁} C (𝒞 := inst)) :
          (a : Yᘁ ⊗ X ⟶ 𝟙_ C) → Yᘁ ⟶ 𝟙_ C ⊗ Xᘁ)
        (@rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ f ≫ ε_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)) :
      @rightDual.{v₁, u₁} _ inst inst_1 Y _ ⟶
        tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@rightDual.{v₁, u₁} _ inst inst_1 X _))

Tactic:
simp
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X X' Y : C
inst_2 : ExactPairing X' Y
i : X ≅ X'
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) Y (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) Y)
    (Y ◁ (η_ X' Y ≫ i.inv ▷ Y) ≫ Iso.inv (α_ Y X Y) ≫ (Y ◁ i.hom ≫ ε_ X' Y) ▷ Y)
    (Y ◁ η_ X' Y ≫ Y ◁ (i.inv ≫ i.hom) ▷ Y ⊗≫ ε_ X' Y ▷ Y)

Tactic:
monoidal
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X X' Y : C
inst_2 : ExactPairing X' Y
i : X ≅ X'
rw :
  Eq.{v₁ + 1} (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) Y (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) Y)
    (Y ◁ η_ X' Y ≫ Y ◁ 𝟙 X' ▷ Y ⊗≫ ε_ X' Y ▷ Y) (Y ◁ η_ X' Y ⊗≫ ε_ X' Y ▷ Y)
⊢ Eq (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) Y (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) Y)
    (Y ◁ η_ X' Y ≫ Y ◁ (i.inv ≫ i.hom) ▷ Y ⊗≫ ε_ X' Y ▷ Y) (Y ◁ η_ X' Y ⊗≫ ε_ X' Y ▷ Y)

Tactic:
rw [Iso.inv_hom_id]

Premises used: CategoryTheory.Iso.inv_hom_id
-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X X' Y : C
inst_2 : ExactPairing X' Y
i : X ≅ X'
⊢ Eq.{v₁ + 1} (α :=
    tensorObj.{v₁, u₁} (𝒞 := inst) Y (tensorUnit.{v₁, u₁} C (𝒞 := inst)) ⟶
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) Y)
    (Y ◁ η_ X' Y ≫ Y ◁ 𝟙 X' ▷ Y ⊗≫ ε_ X' Y ▷ Y) (Y ◁ η_ X' Y ⊗≫ ε_ X' Y ▷ Y)

Tactic:
monoidal