-- 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)
a :
  ∀ (fst : Discrete PUnit.{1}) (x : C),
    Eq.{v₂ + 1} (α :=
      Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠
            Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
          (fst, x) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.prod (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙
            tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
          (fst, x))
      (NatTrans.app
        ((Functor.homEquivOfIsLeftKanExtension
              (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)
              (ExternalProduct.extensionUnitLeft
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
                (φ (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))
              (tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
            (a : (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d) →
              Functor.fromPUnit (𝟙_ V) ⊠ (ι C V D).obj d ⟶
                (Functor.fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj d)
          ((Functor.homEquivOfIsLeftKanExtension
                (@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) _)
                (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))
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
              (a : (ι C V D).obj (𝟙_ D) ⊛ (ι C V D).obj d ⟶ (ι C V D).obj d) →
                (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d)
            (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (λ_ d)))))
        (fst, x))
      (NatTrans.app
        ((Functor.homEquivOfIsLeftKanExtension
              (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)
              (ExternalProduct.extensionUnitLeft
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
                (φ (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))
              (tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
            (a : (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d) →
              Functor.fromPUnit (𝟙_ V) ⊠ (ι C V D).obj d ⟶
                (Functor.fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj d)
          ((Functor.homEquivOfIsLeftKanExtension
                (@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) _)
                (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))
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
              (a : (ι C V D).obj (𝟙_ D) ⊛ (ι C V D).obj d ⟶ (ι C V D).obj d) →
                (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d)
            (Iso.hom
              (DayConvolutionUnit.leftUnitor
                (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)))))
        (fst, x))
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} (Discrete PUnit.{1} × C) V)
      (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
      (Functor.prod (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙
        tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
    ((Functor.homEquivOfIsLeftKanExtension
            (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)
            (ExternalProduct.extensionUnitLeft
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
              (φ (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))
            (tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
          (a : (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d) →
            Functor.fromPUnit (𝟙_ V) ⊠ (ι C V D).obj d ⟶
              (Functor.fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj d)
        ((Functor.homEquivOfIsLeftKanExtension
              (@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) _)
              (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))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
            (a : (ι C V D).obj (𝟙_ D) ⊛ (ι C V D).obj d ⟶ (ι C V D).obj d) →
              (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d)
          (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (λ_ d)))) :
      Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} (Discrete PUnit.{1} × C) V)
        (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠
          Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (Functor.prod (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙
          tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))
    ((Functor.homEquivOfIsLeftKanExtension
            (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)
            (ExternalProduct.extensionUnitLeft
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (tensorUnit.{v₃, u₃} D (𝒞 := inst_4)))
              (φ (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))
            (tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
          (a : (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d) →
            Functor.fromPUnit (𝟙_ V) ⊠ (ι C V D).obj d ⟶
              (Functor.fromPUnit (𝟙_ C)).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj d)
        ((Functor.homEquivOfIsLeftKanExtension
              (@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) _)
              (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))
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d) :
            (a : (ι C V D).obj (𝟙_ D) ⊛ (ι C V D).obj d ⟶ (ι C V D).obj d) →
              (ι C V D).obj (𝟙_ D) ⊠ (ι C V D).obj d ⟶ tensor C ⋙ (ι C V D).obj d)
          (Iso.hom
            (DayConvolutionUnit.leftUnitor
              (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)))) :
      Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} (Discrete PUnit.{1} × C) V)
        (Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1)) ⊠
          Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d)
        (Functor.prod (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) (𝟭 C) ⋙
          tensor C ⋙ Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d))

Tactic:
ext ⟨_, x⟩