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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_6 : DayConvolution U F
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution U G
f : F ⟶ G
hγ :
  ∀ (x : Discrete PUnit.{1} × C),
    Eq (α :=
      (fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F).obj x ⟶
        (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙ tensor C ⋙ G).obj x)
      (NatTrans.app
        (extensionUnitLeft U (φ U) F ≫
          Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
            (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
              (DayConvolution.unit U F)
              (Functor.whiskerLeft (tensor C)
                (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _)
                  (DayConvolution.map (𝟙 U) f) (Iso.hom (leftUnitor U G))))))
        x)
      (NatTrans.app
        (extensionUnitLeft U (φ U) F ≫
          Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
            (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
              (DayConvolution.unit U F) (Functor.whiskerLeft (tensor C) (Iso.hom (leftUnitor U F) ≫ f))))
        x)
⊢ Eq (α :=
    fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F ⟶
      Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙ tensor C ⋙ G)
    (extensionUnitLeft U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
          (DayConvolution.unit U F)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _)
              (DayConvolution.map (𝟙 U) f) (Iso.hom (leftUnitor U G))))))
    (extensionUnitLeft U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
          (DayConvolution.unit U F) (Functor.whiskerLeft (tensor C) (Iso.hom (leftUnitor U F) ≫ f))))

Tactic:
ext
-- 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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_6 : DayConvolution U F
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution U G
f : F ⟶ G
x : Discrete PUnit.{1} × C
⊢ Eq (α :=
    (fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ F).obj x ⟶
      (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙ tensor C ⋙ G).obj x)
    (NatTrans.app
      (extensionUnitLeft U (φ U) F ≫
        Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
          (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
            (DayConvolution.unit U F)
            (Functor.whiskerLeft (tensor C)
              (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U G _)
                (DayConvolution.map (𝟙 U) f) (Iso.hom (leftUnitor U G))))))
      x)
    (NatTrans.app
      (extensionUnitLeft U (φ U) F ≫
        Functor.whiskerLeft (Functor.prod (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C))
          (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 U F _)
            (DayConvolution.unit U F) (Functor.whiskerLeft (tensor C) (Iso.hom (leftUnitor U F) ≫ f))))
      x)

Tactic:
simp [← whisker_exchange_assoc]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange_assoc
-- 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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
x : C
⊢ Eq.{v₂ + 1} (α := F.obj x ⟶ (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).obj x)
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
      (Iso.inv (ρ_ (F.obj x)))
      (CategoryStruct.comp (Y :=
        (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (NatTrans.app
          (Functor.whiskerLeft (prod.rightInverseUnitor C)
            (((F ⊠ U).homEquivOfIsLeftKanExtension (extensionUnitRight U (φ U) F)
                  (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _) :
                (a : F ⊠ U ⟶ tensor C ⋙ F ⊛ U) →
                  F ⊠ fromPUnit (𝟙_ V) ⟶ (𝟭 C).prod (fromPUnit (𝟙_ C)) ⋙ tensor C ⋙ F ⊛ U)
              (((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).homEquivOfIsLeftKanExtension
                    (DayConvolution.unit F U) (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _) :
                  (a : F ⊛ U ⟶ F ⊛ U) → F ⊠ U ⟶ tensor C ⋙ F ⊛ U)
                (𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)))))
          x)
        ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).map (Iso.hom (ρ_ x)))))
    (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
      (Iso.inv (ρ_ (F.obj x)))
      (CategoryStruct.comp (Y := tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (U.obj (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (F.obj x ◁ can)
        (CategoryStruct.comp (Y :=
          (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
          (NatTrans.app (DayConvolution.unit F U) (x, tensorUnit.{v₁, u₁} C (𝒞 := inst)))
          ((@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _).map (Iso.hom (ρ_ x))))))

Tactic:
simp
-- 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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution G U
f : F ⟶ G
hγ :
  Eq (α := F ⊠ U ⟶ tensor C ⋙ G)
    (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
      (DayConvolution.unit F U)
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _) (DayConvolution.map f (𝟙 U))
          (Iso.hom (rightUnitor U G)))))
    (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
      (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f)))
⊢ Eq (α := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _ ⟶ G)
    (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _) (DayConvolution.map f (𝟙 U))
      (Iso.hom (rightUnitor U G)))
    (Iso.hom (rightUnitor U F) ≫ f)

Tactic:
apply Functor.hom_ext_of_isLeftKanExtension _ (DayConvolution.unit _ _) _

Premises used: CategoryTheory.Functor.hom_ext_of_isLeftKanExtension, CategoryTheory.MonoidalCategory.DayConvolution.unit
-- 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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution G U
f : F ⟶ G
hγ :
  Eq (α :=
    F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
      Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ G)
    (extensionUnitRight U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
          (DayConvolution.unit F U)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _)
              (DayConvolution.map f (𝟙 U)) (Iso.hom (rightUnitor U G))))))
    (extensionUnitRight U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
          (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f))))
⊢ Eq (α := F ⊠ U ⟶ tensor C ⋙ G)
    (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
      (DayConvolution.unit F U)
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _) (DayConvolution.map f (𝟙 U))
          (Iso.hom (rightUnitor U G)))))
    (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
      (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f)))

Tactic:
apply Functor.hom_ext_of_isLeftKanExtension _ (extensionUnitRight U (φ U) F) _

Premises used: CategoryTheory.Functor.hom_ext_of_isLeftKanExtension, CategoryTheory.MonoidalCategory.ExternalProduct.extensionUnitRight, CategoryTheory.MonoidalCategory.DayConvolutionUnit.φ
-- 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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution G U
f : F ⟶ G
hγ :
  ∀ (x : C × Discrete PUnit.{1}),
    Eq (α :=
      (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))).obj x ⟶
        (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ G).obj x)
      (NatTrans.app
        (extensionUnitRight U (φ U) F ≫
          Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
            (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
              (DayConvolution.unit F U)
              (Functor.whiskerLeft (tensor C)
                (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _)
                  (DayConvolution.map f (𝟙 U)) (Iso.hom (rightUnitor U G))))))
        x)
      (NatTrans.app
        (extensionUnitRight U (φ U) F ≫
          Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
            (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
              (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f))))
        x)
⊢ Eq (α :=
    F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⟶
      Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ G)
    (extensionUnitRight U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
          (DayConvolution.unit F U)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _)
              (DayConvolution.map f (𝟙 U)) (Iso.hom (rightUnitor U G))))))
    (extensionUnitRight U (φ U) F ≫
      Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
          (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f))))

Tactic:
ext
-- 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
U : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolutionUnit U
F : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_6 : DayConvolution F U
G : Functor.{v₁, v₂, u₁, u₂} C V
inst_7 : DayConvolution G U
f : F ⟶ G
x : C × Discrete PUnit.{1}
⊢ Eq (α :=
    (F ⊠ fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))).obj x ⟶
      (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) ⋙ tensor C ⋙ G).obj x)
    (NatTrans.app
      (extensionUnitRight U (φ U) F ≫
        Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
          (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
            (DayConvolution.unit F U)
            (Functor.whiskerLeft (tensor C)
              (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 G U _)
                (DayConvolution.map f (𝟙 U)) (Iso.hom (rightUnitor U G))))))
      x)
    (NatTrans.app
      (extensionUnitRight U (φ U) F ≫
        Functor.whiskerLeft (Functor.prod (𝟭 C) (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
          (CategoryStruct.comp (Y := tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F U _)
            (DayConvolution.unit F U) (Functor.whiskerLeft (tensor C) (Iso.hom (rightUnitor U F) ≫ f))))
      x)

Tactic:
simp [whisker_exchange_assoc]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange_assoc
-- 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
inst_4 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_6 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
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)
inst_8 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        d)
      (tensorRight v)
F G U : Functor.{v₁, v₂, u₁, u₂} C V
inst_9 : DayConvolutionUnit U
inst_10 : DayConvolution F U
inst_11 : DayConvolution U G
inst_12 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 U G _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F U _) G
inst_14 : DayConvolution F G
hγ :
  Eq (α := @convolution _ _ _ _ inst_2 inst_3 F U _ ⊠ G ⟶ tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F G _)
    (CategoryStruct.comp (Y :=
      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
      (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _)
          (Iso.hom (associator F U G)) (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G))))))
    (CategoryStruct.comp (Y :=
      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
      (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
      (Functor.whiskerLeft (tensor C) (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G))))
⊢ Eq (α :=
    @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _ ⟶
      @convolution _ _ _ _ inst_2 inst_3 F G _)
    (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _)
      (Iso.hom (associator F U G)) (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G))))
    (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G))

Tactic:
apply Functor.hom_ext_of_isLeftKanExtension _ (DayConvolution.unit _ _) _

Premises used: CategoryTheory.Functor.hom_ext_of_isLeftKanExtension, CategoryTheory.MonoidalCategory.DayConvolution.unit
-- 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
inst_4 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_6 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
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)
inst_8 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        d)
      (tensorRight v)
F G U : Functor.{v₁, v₂, u₁, u₂} C V
inst_9 : DayConvolutionUnit U
inst_10 : DayConvolution F U
inst_11 : DayConvolution U G
inst_12 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 U G _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F U _) G
inst_14 : DayConvolution F G
⊢ ((F ⊠ U) ⊠ G).IsLeftKanExtension (L :=
    Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C)) (F :=
    (F ⊠ Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) ⊠ G)
    (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G)

Tactic:
have : (F ⊠ U) ⊠ G |>.IsLeftKanExtension
      (α := extensionUnitLeft (F ⊠ U) (extensionUnitRight U (DayConvolutionUnit.φ U) F) G) :=
    isPointwiseLeftKanExtensionExtensionUnitLeft (F ⊠ U) _ G
      (isPointwiseLeftKanExtensionExtensionUnitRight U (DayConvolutionUnit.φ U) F <|
        DayConvolutionUnit.isPointwiseLeftKanExtensionCan (F := U)) |>.isLeftKanExtension
-- 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
inst_4 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_6 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
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)
inst_8 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        d)
      (tensorRight v)
F G U : Functor.{v₁, v₂, u₁, u₂} C V
inst_9 : DayConvolutionUnit U
inst_10 : DayConvolution F U
inst_11 : DayConvolution U G
inst_12 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 U G _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F U _) G
inst_14 : DayConvolution F G
this :
  ((F ⊠ U) ⊠ G).IsLeftKanExtension (L :=
    Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C)) (F :=
    (F ⊠ Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) ⊠ G)
    (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G)
x : (C × Discrete PUnit.{1}) × C
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
        (tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) (G.obj x.2)) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 F G _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst))) x.2))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
        (G.obj (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2)))
      (F.obj x.1.1 ◁ (Iso.hom (λ_ (G.obj x.2)) ≫ G.map (Iso.inv (λ_ x.2))))
      (NatTrans.app (unit F G) (x.1.1, tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2) ≫
        (@convolution _ _ _ _ inst_2 inst_3 F G _).map (Iso.inv (α_ x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)) x.2))))
    (Iso.inv (α_ (F.obj x.1.1) (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) (G.obj x.2)) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          (F.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (G.obj x.2))
        ((Iso.hom (ρ_ (F.obj x.1.1)) ≫ F.map (Iso.inv (ρ_ x.1.1))) ▷ G.obj x.2)
        (NatTrans.app (unit F G) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 (tensorUnit.{v₁, u₁} C (𝒞 := inst)), x.2)))

Tactic:
simp [← Functor.map_comp]

Premises used: CategoryTheory.Functor.map_comp