-- 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)
a b c d : D
this_1 : DayConvolution.{v₁, v₂, u₁, u₂}
  (@DayConvolution.convolution _ _ _ _ inst_6 inst_7
    (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) b) _)
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) c) _)
  (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d) :=
  LawfulDayConvolutionMonoidalCategoryStruct.convolution C V D
    (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a b) c) d
this_2 : DayConvolution.{v₁, v₂, u₁, u₂}
  (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a)
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) b) _)
  (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) c)
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d) _) :=
  LawfulDayConvolutionMonoidalCategoryStruct.convolution C V D (tensorObj.{v₃, u₃} (𝒞 := inst_8) a b)
    (tensorObj.{v₃, u₃} (𝒞 := inst_8) c d)
this_3 : DayConvolution.{v₁, v₂, u₁, u₂}
  (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a)
    (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) b)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) c) _)
    _)
  (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d) :=
  LawfulDayConvolutionMonoidalCategoryStruct.convolution C V D
    (tensorObj.{v₃, u₃} (𝒞 := inst_8) a (tensorObj.{v₃, u₃} (𝒞 := inst_8) b c)) d
this_4 : DayConvolution.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a)
  (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) b)
    (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) c)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d) _)
    _) :=
  LawfulDayConvolutionMonoidalCategoryStruct.convolution C V D a
    (tensorObj.{v₃, u₃} (𝒞 := inst_8) b (tensorObj.{v₃, u₃} (𝒞 := inst_8) c d))
this : DayConvolution.{v₁, v₂, u₁, u₂} (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a)
  (@DayConvolution.convolution _ _ _ _ inst_6 inst_7
    (@DayConvolution.convolution _ _ _ _ inst_6 inst_7 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) b)
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) c) _)
    (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d) _) :=
  LawfulDayConvolutionMonoidalCategoryStruct.convolution C V D a
    (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) b c) d)
⊢ 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) (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a b) c)
          d))
      (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
        (tensorObj.{v₃, u₃} (𝒞 := inst_8) a
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) b (tensorObj.{v₃, u₃} (𝒞 := inst_8) c d)))))
    (CategoryStruct.comp (Y :=
        Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a (tensorObj.{v₃, u₃} (𝒞 := inst_8) b c))
            d))
        (map
          (Iso.hom
            (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) b)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) c)))
          (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d)))
        (CategoryStruct.comp (Y :=
          Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
            (tensorObj.{v₃, u₃} (𝒞 := inst_8) a
              (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) b c) d)))
          (Iso.hom
            (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a)
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) b c))
              (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d)))
          (map (𝟙 (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a))
            (Iso.hom
              (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) b)
                (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) c)
                (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d))))) :
      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) (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a b) c)
            d))
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) a
            (tensorObj.{v₃, u₃} (𝒞 := inst_8) b (tensorObj.{v₃, u₃} (𝒞 := inst_8) c d)))))
    (CategoryStruct.comp (Y :=
        Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a b)
            (tensorObj.{v₃, u₃} (𝒞 := inst_8) c d)))
        (Iso.hom
          (DayConvolution.associator
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a b))
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) c)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) d)))
        (Iso.hom
          (DayConvolution.associator (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) a)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) b)
            (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _) (tensorObj.{v₃, u₃} (𝒞 := inst_8) c d)))) :
      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) (tensorObj.{v₃, u₃} (𝒞 := inst_8) (tensorObj.{v₃, u₃} (𝒞 := inst_8) a b) c)
            d))
        (Functor.obj (@ι C _ V _ inst_6 inst_7 D _ inst_9 _)
          (tensorObj.{v₃, u₃} (𝒞 := inst_8) a
            (tensorObj.{v₃, u₃} (𝒞 := inst_8) b (tensorObj.{v₃, u₃} (𝒞 := inst_8) c d)))))

Tactic:
exact DayConvolution.pentagon
        (ι C V D |>.obj a) (ι C V D |>.obj b) (ι C V D |>.obj c) (ι C V D |>.obj d)

Premises used: CategoryTheory.MonoidalCategory.DayConvolution.pentagon, CategoryTheory.MonoidalCategory.LawfulDayConvolutionMonoidalCategoryStruct.ι, CategoryTheory.Functor.obj