awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
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 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
inst_6 : DayConvolution F H
⊢ CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _)
      (DayConvolution.map (𝟙 F) ℌ.coev_app) ℌ.ev_app =
    𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)
have a : Eq (α :=
  (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙ Functor.obj coyoneda (op (F ⊠ G))).obj
    (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _))
  ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
        (a : F ⊛ G ⟶ F ⊛ G) →
          ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙ coyoneda.obj (op (F ⊠ G))).obj (F ⊛ G))
      (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _)
        (DayConvolution.map (𝟙 F) ℌ.coev_app) ℌ.ev_app) :
    (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙ Functor.obj coyoneda (op (F ⊠ G))).obj
      (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _))
  ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
        (a : F ⊛ G ⟶ F ⊛ G) →
          ((Functor.whiskeringLeft (C × C) C V).obj (tensor C) ⋙ coyoneda.obj (op (F ⊠ G))).obj (F ⊛ G))
      (𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)) :
    (Functor.obj (Functor.whiskeringLeft (C × C) C V) (tensor C) ⋙ Functor.obj coyoneda (op (F ⊠ G))).obj
      (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _))
Tactic: apply DayConvolution.corepresentableBy F G |>.homEquiv.injective
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
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 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
inst_6 : DayConvolution F H
⊢ Eq (α := F ⊠ G ⟶ tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)
    ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
          (a : F ⊛ G ⟶ F ⊛ G) → F ⊠ G ⟶ tensor C ⋙ F ⊛ G)
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _)
          (DayConvolution.map (𝟙 F) ℌ.coev_app) ℌ.ev_app) :
      F ⊠ G ⟶ tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)
    ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
          (a : F ⊛ G ⟶ F ⊛ G) → F ⊠ G ⟶ tensor C ⋙ F ⊛ G)
        (𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)) :
      F ⊠ G ⟶ tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)
have a : ∀ (x y : C),
  Eq.{v₂ + 1} (α :=
    (F ⊠ G).obj (x, y) ⟶ (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (x, y))
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
          (a : F ⊛ G ⟶ F ⊛ G) → F ⊠ G ⟶ tensor C ⋙ F ⊛ G)
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _)
          (DayConvolution.map (𝟙 F) ℌ.coev_app) ℌ.ev_app))
      (x, y))
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
          (a : F ⊛ G ⟶ F ⊛ G) → F ⊠ G ⟶ tensor C ⋙ F ⊛ G)
        (𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)))
      (x, y))
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
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 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F G
ℌ : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _) H
inst_6 : DayConvolution F H
x y : C
⊢ Eq.{v₂ + 1} (α :=
    (F ⊠ G).obj (x, y) ⟶ (tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (x, y))
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
          (a : F ⊛ G ⟶ F ⊛ G) → F ⊠ G ⟶ tensor C ⋙ F ⊛ G)
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _)
          (DayConvolution.map (𝟙 F) ℌ.coev_app) ℌ.ev_app))
      (x, y))
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
          (a : F ⊛ G ⟶ F ⊛ G) → F ⊠ G ⟶ tensor C ⋙ F ⊛ G)
        (𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)))
      (x, y))
have a : Eq.{v₂ + 1} (α :=
  (F, G).2.obj (x, y).2 ⟶
    Functor.obj.{v₂, v₂, u₂, u₂} (ihom ((F, G).1.obj (x, y).1))
      ((tensor C ⋙ @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _).obj (x, y)))
  (MonoidalClosed.curry
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
          (a : F ⊛ G ⟶ F ⊛ G) → F ⊠ G ⟶ tensor C ⋙ F ⊛ G)
        (CategoryStruct.comp (Y := @DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _)
          (DayConvolution.map (𝟙 F) ℌ.coev_app) ℌ.ev_app))
      (x, y)))
  (MonoidalClosed.curry
    (NatTrans.app
      ((Functor.CorepresentableBy.homEquiv (DayConvolution.corepresentableBy F G) :
          (a : F ⊛ G ⟶ F ⊛ G) → F ⊠ G ⟶ tensor C ⋙ F ⊛ G)
        (𝟙 (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F G _)))
      (x, y)))
Tactic: apply MonoidalClosed.curry_injective
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
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 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
H' : Functor.{v₁, v₂, u₁, u₂} C V
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _) H'
⊢ ℌ'.coev_app ≫ ℌ'.map ℌ.ev_app ℌ = 𝟙 H
have w : ∀ (c : C), (ℌ'.coev_app ≫ ℌ'.map ℌ.ev_app ℌ).app c = NatTrans.app (𝟙 H) c
Tactic: ext c
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
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 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
H' : Functor.{v₁, v₂, u₁, u₂} C V
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _) H'
c : C
⊢ (ℌ'.coev_app ≫ ℌ'.map ℌ.ev_app ℌ).app c = NatTrans.app (𝟙 H) c
have w : ∀ (j : (multicospanShapeEnd.{v₁, u₁} C).L),
  Eq (α := H.obj c ⟶ (multicospanIndexEnd (((dayConvolutionInternalHomDiagramFunctor F).obj G).obj c)).left j)
    ((ℌ'.coev_app ≫ ℌ'.map ℌ.ev_app ℌ).app c ≫ Multifork.ι (Wedge.mk (H.obj c) (ℌ.π c) sorry) j)
    (NatTrans.app (𝟙 H) c ≫ Multifork.ι (Wedge.mk (H.obj c) (ℌ.π c) sorry) j)
Tactic: apply Wedge.IsLimit.hom_ext <| ℌ.isLimitWedge c
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.Closed (goal)
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 : MonoidalClosed.{v₂, u₂} V
F H G : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution F H
ℌ : DayConvolutionInternalHom F G H
H' : Functor.{v₁, v₂, u₁, u₂} C V
ℌ' : DayConvolutionInternalHom F (@DayConvolution.convolution _ _ _ _ inst_2 inst_3 F H _) H'
c : C
j : (multicospanShapeEnd.{v₁, u₁} C).L
⊢ Eq (α := H.obj c ⟶ (multicospanIndexEnd (((dayConvolutionInternalHomDiagramFunctor F).obj G).obj c)).left j)
    ((ℌ'.coev_app ≫ ℌ'.map ℌ.ev_app ℌ).app c ≫ Multifork.ι (Wedge.mk (H.obj c) (ℌ.π c) ⋯) j)
    (NatTrans.app (𝟙 H) c ≫ Multifork.ι (Wedge.mk (H.obj c) (ℌ.π c) ⋯) j)
have w : Eq (α := tensorObj.{v₂, u₂} (𝒞 := inst_1) (unop (F.op.obj (op j))) (H.obj c) ⟶ (tensorRight c ⋙ G).obj j)
  (MonoidalClosed.uncurry ((ℌ'.coev_app ≫ ℌ'.map ℌ.ev_app ℌ).app c ≫ Multifork.ι (Wedge.mk (H.obj c) (ℌ.π c) sorry) j))
  (MonoidalClosed.uncurry (NatTrans.app (𝟙 H) c ≫ Multifork.ι (Wedge.mk (H.obj c) (ℌ.π c) sorry) j))
Tactic: apply MonoidalClosed.uncurry_injective
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor (goal)
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
F G : DayFunctor.{v₁, v₂, u₁, u₂} C V
α β : F ⟶ G
h : α.natTrans = β.natTrans
⊢ α = β
have mk : ∀ (natTrans : F.functor ⟶ G.functor) (h : { natTrans := natTrans : F.Hom G }.natTrans = β.natTrans),
  { natTrans := natTrans : F.Hom G } = β
Tactic: cases α
Cleared hypotheses: α, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor (goal)
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
F G : DayFunctor.{v₁, v₂, u₁, u₂} C V
β : F ⟶ G
natTrans : F.functor ⟶ G.functor
h : { natTrans := natTrans : F.Hom G }.natTrans = β.natTrans
⊢ { natTrans := natTrans : F.Hom G } = β
have mk : ∀ (natTrans_1 : F.functor ⟶ G.functor)
  (h : { natTrans := natTrans : F.Hom G }.natTrans = { natTrans := natTrans_1 }.natTrans),
  { natTrans := natTrans : F.Hom G } = { natTrans := natTrans_1 }
Tactic: cases β
Cleared hypotheses: β, h
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor (goal)
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
hasDayConvolution : ∀ (F G : Functor.{v₁, v₂, u₁, u₂} C V), Functor.HasPointwiseLeftKanExtension (tensor C) (F ⊠ G)
hasDayConvolutionUnit :
  Functor.HasPointwiseLeftKanExtension.{0, 0, v₁, u₁, v₂, u₂} (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
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)
inst_9 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
F G H : DayFunctor.{v₁, v₂, u₁, u₂} C V
α β : tensorObj (𝒞 := instCategory) F G ⟶ H
h :
  ∀ (x y : C),
    Eq (α := (F.functor ⊠ G.functor).obj (x, y) ⟶ H.functor.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
      (NatTrans.app (F.η G) (x, y) ≫ α.natTrans.app (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
      (NatTrans.app (F.η G) (x, y) ≫ β.natTrans.app (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
⊢ α = β
have h₁ : α.natTrans = β.natTrans
Tactic: ext : 1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor (goal)
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
hasDayConvolution : ∀ (F G : Functor.{v₁, v₂, u₁, u₂} C V), Functor.HasPointwiseLeftKanExtension (tensor C) (F ⊠ G)
hasDayConvolutionUnit :
  Functor.HasPointwiseLeftKanExtension.{0, 0, v₁, u₁, v₂, u₂} (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
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)
inst_9 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
F G H : DayFunctor.{v₁, v₂, u₁, u₂} C V
α β : tensorObj (𝒞 := instCategory) F G ⟶ H
h :
  ∀ (x y : C),
    Eq (α := (F.functor ⊠ G.functor).obj (x, y) ⟶ H.functor.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
      (NatTrans.app (F.η G) (x, y) ≫ α.natTrans.app (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
      (NatTrans.app (F.η G) (x, y) ≫ β.natTrans.app (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
⊢ α.natTrans = β.natTrans
have h₁ : F.η G ≫ Functor.whiskerLeft (tensor C) α.natTrans = F.η G ≫ Functor.whiskerLeft (tensor C) β.natTrans
Tactic: apply Functor.hom_ext_of_isLeftKanExtension
    (F ⊗ G).functor (η F G) _
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor (goal)
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
hasDayConvolution : ∀ (F G : Functor.{v₁, v₂, u₁, u₂} C V), Functor.HasPointwiseLeftKanExtension (tensor C) (F ⊠ G)
hasDayConvolutionUnit :
  Functor.HasPointwiseLeftKanExtension.{0, 0, v₁, u₁, v₂, u₂} (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
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)
inst_9 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
F G H : DayFunctor.{v₁, v₂, u₁, u₂} C V
α β : tensorObj (𝒞 := instCategory) F G ⟶ H
h :
  ∀ (x y : C),
    Eq (α := (F.functor ⊠ G.functor).obj (x, y) ⟶ H.functor.obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
      (NatTrans.app (F.η G) (x, y) ≫ α.natTrans.app (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
      (NatTrans.app (F.η G) (x, y) ≫ β.natTrans.app (tensorObj.{v₁, u₁} (𝒞 := inst) x y))
⊢ F.η G ≫ Functor.whiskerLeft (tensor C) α.natTrans = F.η G ≫ Functor.whiskerLeft (tensor C) β.natTrans
have h₁ : ∀ (x y : C),
  Eq (α := (F.functor ⊠ G.functor).obj (x, y) ⟶ (tensor C ⋙ H.functor).obj (x, y))
    ((F.η G ≫ Functor.whiskerLeft (tensor C) α.natTrans).app (x, y))
    ((F.η G ≫ Functor.whiskerLeft (tensor C) β.natTrans).app (x, y))
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor (goal)
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
hasDayConvolution : ∀ (F G : Functor.{v₁, v₂, u₁, u₂} C V), Functor.HasPointwiseLeftKanExtension (tensor C) (F ⊠ G)
hasDayConvolutionUnit :
  Functor.HasPointwiseLeftKanExtension.{0, 0, v₁, u₁, v₂, u₂} (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
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)
inst_9 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
F : DayFunctor.{v₁, v₂, u₁, u₂} C V
α β : (𝟙_ (DayFunctor.{v₁, v₂, u₁, u₂} C V)) (𝒞 := instCategory) ⟶ F
h :
  ν C V ≫ α.natTrans.app (tensorUnit.{v₁, u₁} C (𝒞 := inst)) =
    ν C V ≫ β.natTrans.app (tensorUnit.{v₁, u₁} C (𝒞 := inst))
⊢ α = β
have h₁ : α.natTrans = β.natTrans
Tactic: ext1
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor (goal)
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
hasDayConvolution : ∀ (F G : Functor.{v₁, v₂, u₁, u₂} C V), Functor.HasPointwiseLeftKanExtension (tensor C) (F ⊠ G)
hasDayConvolutionUnit :
  Functor.HasPointwiseLeftKanExtension.{0, 0, v₁, u₁, v₂, u₂} (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
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)
inst_9 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
F : DayFunctor.{v₁, v₂, u₁, u₂} C V
α β : (𝟙_ (DayFunctor.{v₁, v₂, u₁, u₂} C V)) (𝒞 := instCategory) ⟶ F
h :
  ν C V ≫ α.natTrans.app (tensorUnit.{v₁, u₁} C (𝒞 := inst)) =
    ν C V ≫ β.natTrans.app (tensorUnit.{v₁, u₁} C (𝒞 := inst))
⊢ α.natTrans = β.natTrans
have h₁ : νNatTrans C V ≫ Functor.whiskerLeft (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) α.natTrans =
  νNatTrans C V ≫ Functor.whiskerLeft (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) β.natTrans
Tactic: apply Functor.hom_ext_of_isLeftKanExtension
    (𝟙_ (C ⊛⥤ V)).functor (νNatTrans C V)
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution.DayFunctor (goal)
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
hasDayConvolution : ∀ (F G : Functor.{v₁, v₂, u₁, u₂} C V), Functor.HasPointwiseLeftKanExtension (tensor C) (F ⊠ G)
hasDayConvolutionUnit :
  Functor.HasPointwiseLeftKanExtension.{0, 0, v₁, u₁, v₂, u₂} (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
    (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
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)
inst_9 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
F : DayFunctor.{v₁, v₂, u₁, u₂} C V
α β : (𝟙_ (DayFunctor.{v₁, v₂, u₁, u₂} C V)) (𝒞 := instCategory) ⟶ F
h :
  ν C V ≫ α.natTrans.app (tensorUnit.{v₁, u₁} C (𝒞 := inst)) =
    ν C V ≫ β.natTrans.app (tensorUnit.{v₁, u₁} C (𝒞 := inst))
⊢ νNatTrans C V ≫ Functor.whiskerLeft (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) α.natTrans =
    νNatTrans C V ≫ Functor.whiskerLeft (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) β.natTrans
have h₁ : ∀ (x : Discrete PUnit.{1}),
  (νNatTrans C V ≫ Functor.whiskerLeft (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) α.natTrans).app x =
    (νNatTrans C V ≫ Functor.whiskerLeft (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) β.natTrans).app x
Tactic: ext
-- Lean 4 proof step from Mathlib.CategoryTheory.Monoidal.DayConvolution (goal)
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
c : C
v : V
g h : U.obj c ⟶ v
e :
  ∀ (f : tensorUnit.{v₁, u₁} C (𝒞 := inst) ⟶ c),
    Eq (α := tensorUnit.{v₂, u₂} V (𝒞 := inst_1) ⟶ v) (can ≫ U.map f ≫ g) (can ≫ U.map f ≫ h)
⊢ g = h
have apply : ∀ (j : CostructuredArrow (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) c),
  Eq (α :=
    obj.{0, v₂, v₁, u₂}
        (CostructuredArrow.proj (fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) c ⋙
          fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)))
        j ⟶
      v)
    (NatTrans.app
        (Limits.Cocone.ι
          (LeftExtension.coconeAt (LeftExtension.mk U { app := fun (x : Discrete PUnit.{1}) ↦ can, naturality := sorry })
            c))
        j ≫
      g)
    (NatTrans.app
        (Limits.Cocone.ι
          (LeftExtension.coconeAt (LeftExtension.mk U { app := fun (x : Discrete PUnit.{1}) ↦ can, naturality := sorry })
            c))
        j ≫
      h)
Tactic: apply (isPointwiseLeftKanExtensionCan c).hom_ext