awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d₁ d₂ d₁' d₂' : D
f : d₁ ⟶ d₂
f' : d₁' ⟶ d₂'
x y : C
⊢ Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁) x)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁') y) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂'))
        (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
    (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁'))
          (x, y) ≫
        NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (f ⊗ₘ f'))
          (tensorObj.{v₁, u₁} (𝒞 := inst) x y) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁) x)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁') y) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂'))
          (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
    ((NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) f) x ⊗ₘ
          NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) f') y) ≫
        NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₂)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₂'))
          (x, y) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁) x)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁') y) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₂)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₂') _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) x y))

Tactic:
convolutionExtensionUnit_comp_ι_map_tensorHom_app C V _ _ _ _

Premises used: CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.convolutionExtensionUnit_comp_ι_map_tensorHom_app
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d₁ d₂ d₁' d₂' : D
f : d₁ ⟶ d₂
f' : d₁' ⟶ d₂'
x y : C
⊢ Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁) x)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁') y) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂'))
        (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
    (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁'))
          (x, y) ≫
        NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (f ⊗ₘ f'))
          (tensorObj.{v₁, u₁} (𝒞 := inst) x y) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁) x)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁') y) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorObj.{v₃, u₃} (𝒞 := inst_4) d₂ d₂'))
          (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
    ((NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) f) x ⊗ₘ
          NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) f') y) ≫
        NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₂)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₂'))
          (x, y) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁) x)
          (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₁') y) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₂)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d₂') _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) x y))

Tactic:
exact convolutionExtensionUnit_comp_ι_map_tensorHom_app C V _ _ _ _

Premises used: CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.convolutionExtensionUnit_comp_ι_map_tensorHom_app
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d : D
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
fst : Discrete PUnit.{1}
x : C
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
          (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x))
      (can ▷ Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
      (CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
        (NatTrans.app
          (DayConvolution.unit
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
          (tensorUnit.{v₁, u₁} C (𝒞 := inst), x))
        (NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (λ_ d)))
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))))
    (Iso.hom (λ_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)) ≫
      Functor.map (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) (Iso.inv (λ_ x)))

Tactic:
leftUnitor_hom_unit_app V d x

Premises used: CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.leftUnitor_hom_unit_app
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d : D
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
fst : Discrete PUnit.{1}
x : C
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
          (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x))
      (can ▷ Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
      (CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))
        (NatTrans.app
          (DayConvolution.unit
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
          (tensorUnit.{v₁, u₁} C (𝒞 := inst), x))
        (NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (λ_ d)))
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x))))
    (Iso.hom (λ_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)) ≫
      Functor.map (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) (Iso.inv (λ_ x)))

Tactic:
exact leftUnitor_hom_unit_app V d x

Premises used: CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.leftUnitor_hom_unit_app
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d : D
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
x : C
snd : Discrete PUnit.{1}
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
        (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
          (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
      (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x ◁ can)
      (CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4))) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4))))
          (x, tensorUnit.{v₁, u₁} C (𝒞 := inst)))
        (NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (ρ_ d)))
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))))
    (Iso.hom (ρ_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)) ≫
      Functor.map (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) (Iso.inv (ρ_ x)))

Tactic:
rightUnitor_hom_unit_app V d x

Premises used: CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.rightUnitor_hom_unit_app
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
V : Type u₂
inst_1 : Category.{v₂, u₂} V
inst_2 : MonoidalCategory.{v₁, u₁} C
inst_3 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_4 : Category.{v₃, u₃} D
inst_5 : MonoidalCategoryStruct.{v₃, u₃} D
inst_6 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
d : D
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
x : C
snd : Discrete PUnit.{1}
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
        (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
          (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
      (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x ◁ can)
      (CategoryStruct.comp (Y :=
        Functor.obj.{v₁, v₂, u₁, u₂}
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4))) _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (NatTrans.app
          (DayConvolution.unit (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
            (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4))))
          (x, tensorUnit.{v₁, u₁} C (𝒞 := inst)))
        (NatTrans.app (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (ρ_ d)))
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))))
    (Iso.hom (ρ_ (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) x)) ≫
      Functor.map (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) (Iso.inv (ρ_ x)))

Tactic:
exact rightUnitor_hom_unit_app V d x

Premises used: CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.rightUnitor_hom_unit_app
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : MonoidalCategoryStruct.{v₃, u₃} D
inst_10 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_12 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorLeft v)
inst_14 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorRight v)
inst_15 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
        d)
      (tensorRight v)
inst_16 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
x y : D
a :
  Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 x ⊗ₘ 𝟙 y))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
⊢ 𝟙 x ⊗ₘ 𝟙 y = 𝟙 (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)

Tactic:
apply Functor.Faithful.map_injective (F := ι C V D)

Premises used: CategoryTheory.Functor.Faithful.map_injective
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : MonoidalCategoryStruct.{v₃, u₃} D
inst_10 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_12 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorLeft v)
inst_14 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorRight v)
inst_15 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
        d)
      (tensorRight v)
inst_16 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
x y : D
a :
  Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
    (map (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x))
        (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)) :
      Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
        (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) _)
        (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) _))
    (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} C V)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 x ⊗ₘ 𝟙 y))
    (Functor.map (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (𝟙 (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y)))

Tactic:
simp only [ι_map_tensorHom_hom_eq_tensorHom, Functor.map_id]

Premises used: CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.ι_map_tensorHom_hom_eq_tensorHom, CategoryTheory.Functor.map_id
-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution
Goal:
C_1 : Type u₁
inst : Category.{v₁, u₁} C_1
V_1 : Type u₂
inst_1 : Category.{v₂, u₂} V_1
inst_2 : MonoidalCategory.{v₁, u₁} C_1
inst_3 : MonoidalCategory.{v₂, u₂} V_1
C : Type u₁
inst_4 : Category.{v₁, u₁} C
V : Type u₂
inst_5 : Category.{v₂, u₂} V
inst_6 : MonoidalCategory.{v₁, u₁} C
inst_7 : MonoidalCategory.{v₂, u₂} V
D : Type u₃
inst_8 : Category.{v₃, u₃} D
inst_9 : MonoidalCategoryStruct.{v₃, u₃} D
inst_10 : LawfulDayConvolutionMonoidalCategoryStruct.{v₁, v₂, v₃, u₁, u₂, u₃} C V D
inst_11 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_12 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_13 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorLeft v)
inst_14 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))) d) (tensorRight v)
inst_15 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst_4))))
        d)
      (tensorRight v)
inst_16 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
x y : D
fst snd : C
⊢ Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_5)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x) fst)
        (Functor.obj.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y) snd) ⟶
      Functor.obj.{v₁, v₂, u₁, u₂}
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))
        (tensorObj.{v₁, u₁} (𝒞 := inst_4) fst snd))
    (NatTrans.app
        (unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y))
        (fst, snd) :
      Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x ⊠
            Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y)
          (fst, snd) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (tensor C ⋙ Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))
          (fst, snd))
    (NatTrans.app
        (unit (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) x)
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) y))
        (fst, snd) ≫
      𝟙
        (Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) x y))
          (tensorObj.{v₁, u₁} (𝒞 := inst_4) fst snd)))

Tactic:
simp