-- 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' d'' : D
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_8 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
a :
  ∀ (z x y : C),
    Eq.{v₂ + 1} (α :=
      Functor.obj.{v₁, v₂, u₁, u₂}
          ((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') ⊠
            Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
          ((x, y), z) ⟶
        Functor.obj.{v₁, v₂, u₁, u₂}
          (Functor.prod (tensor C) (𝟭 C) ⋙
            tensor C ⋙
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
                (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d'')))
          ((x, y), z))
      (NatTrans.app
        ((Functor.homEquivOfIsLeftKanExtension
              (@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 _) d') _ ⊠
                Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
              (ExternalProduct.extensionUnitLeft
                (@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 _) d') _)
                (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 _) d'))
                (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 _)
                  (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))) :
            (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
              ((ι C V D).obj d ⊠ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶
                (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
          ((Functor.CorepresentableBy.homEquiv
                (corepresentableBy
                  (@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 _) d') _)
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')) :
              (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊛ (ι C V D).obj d'' ⟶ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
                ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
            (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (α_ d d' d'')))))
        ((x, y), z))
      (NatTrans.app
        ((Functor.homEquivOfIsLeftKanExtension
              (@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 _) d') _ ⊠
                Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
              (ExternalProduct.extensionUnitLeft
                (@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 _) d') _)
                (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 _) d'))
                (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 _)
                  (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))) :
            (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
              ((ι C V D).obj d ⊠ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶
                (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
          ((Functor.CorepresentableBy.homEquiv
                (corepresentableBy
                  (@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 _) d') _)
                  (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')) :
              (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊛ (ι C V D).obj d'' ⟶ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
                ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
            (Iso.hom
              (DayConvolution.associator (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')
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')))))
        ((x, y), z))
⊢ Eq (α :=
    Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} ((C × C) × C) V)
      ((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') ⊠
        Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
      (Functor.prod (tensor C) (𝟭 C) ⋙
        tensor C ⋙
          Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
            (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))))
    ((Functor.homEquivOfIsLeftKanExtension
            (@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 _) d') _ ⊠
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
            (ExternalProduct.extensionUnitLeft
              (@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 _) d') _)
              (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 _) d'))
              (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 _)
                (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))) :
          (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
            ((ι C V D).obj d ⊠ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶
              (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
        ((Functor.CorepresentableBy.homEquiv
              (corepresentableBy
                (@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 _) d') _)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')) :
            (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊛ (ι C V D).obj d'' ⟶ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
              ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
          (Functor.map (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) (Iso.hom (α_ d d' d'')))) :
      Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} ((C × C) × C) V)
        ((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') ⊠
          Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
        (Functor.prod (tensor C) (𝟭 C) ⋙
          tensor C ⋙
            Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
              (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))))
    ((Functor.homEquivOfIsLeftKanExtension
            (@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 _) d') _ ⊠
              Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
            (ExternalProduct.extensionUnitLeft
              (@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 _) d') _)
              (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 _) d'))
              (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 _)
                (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))) :
          (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
            ((ι C V D).obj d ⊠ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶
              (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
        ((Functor.CorepresentableBy.homEquiv
              (corepresentableBy
                (@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 _) d') _)
                (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')) :
            (a : ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊛ (ι C V D).obj d'' ⟶ (ι C V D).obj (d ⊗ d' ⊗ d'')) →
              ((ι C V D).obj d ⊛ (ι C V D).obj d') ⊠ (ι C V D).obj d'' ⟶ tensor C ⋙ (ι C V D).obj (d ⊗ d' ⊗ d''))
          (Iso.hom
            (DayConvolution.associator (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')
              (Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')))) :
      Quiver.Hom (V := Functor.{v₁, v₂, u₁, u₂} ((C × C) × C) V)
        ((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') ⊠
          Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _) d'')
        (Functor.prod (tensor C) (𝟭 C) ⋙
          tensor C ⋙
            Functor.obj (@ι C _ V _ inst_2 inst_3 D _ inst_5 _)
              (tensorObj.{v₃, u₃} (𝒞 := inst_4) d (tensorObj.{v₃, u₃} (𝒞 := inst_4) d' d''))))

Tactic:
ext ⟨⟨x, y⟩, z⟩