-- 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
F G : Functor.{v₁, v₂, u₁, u₂} C V
inst_4 : DayConvolution F G
H : Functor.{v₁, v₂, u₁, u₂} C V
inst_5 : DayConvolution G H
inst_6 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 G H _)
inst_7 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F G _) H
inst_8 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_9 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
F' G' H' : Functor.{v₁, v₂, u₁, u₂} C V
inst_10 : DayConvolution F' G'
inst_11 : DayConvolution G' H'
inst_12 : DayConvolution F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F' G' _) H'
f : F ⟶ F'
g : G ⟶ G'
h : H ⟶ H'
a :
  ∀ (x : (C × C) × C),
    Eq.{v₂ + 1} (α :=
      ((F ⊠ G) ⊠ H).obj x ⟶
        (Functor.prod (tensor C) (𝟭 C) ⋙
              tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _).obj
          x)
      (NatTrans.app
        (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
              (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
              (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _) :
            (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H') → (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F' ⊛ G' ⊛ H')
          ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
              (a : (F ⊛ G) ⊛ H ⟶ F' ⊛ G' ⊛ H') → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H')
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F' G' _) H' _) (map (map f g) h)
              (Iso.hom (associator F' G' H')))))
        x)
      (NatTrans.app
        (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
              (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
              (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _) :
            (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H') → (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F' ⊛ G' ⊛ H')
          ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
              (a : (F ⊛ G) ⊛ H ⟶ F' ⊛ G' ⊛ H') → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H')
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
              (Iso.hom (associator F G H)) (map f (map g h)))))
        x)
⊢ Eq (α :=
    (F ⊠ G) ⊠ H ⟶
      Functor.prod (tensor C) (𝟭 C) ⋙
        tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _)
    (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
            (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
            (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _) :
          (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H') → (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F' ⊛ G' ⊛ H')
        ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
            (a : (F ⊛ G) ⊛ H ⟶ F' ⊛ G' ⊛ H') → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H')
          (CategoryStruct.comp (Y :=
            @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F' G' _) H' _) (map (map f g) h)
            (Iso.hom (associator F' G' H')))) :
      (F ⊠ G) ⊠ H ⟶
        Functor.prod (tensor C) (𝟭 C) ⋙
          tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _)
    (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
            (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
            (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _) :
          (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H') → (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F' ⊛ G' ⊛ H')
        ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
            (a : (F ⊛ G) ⊛ H ⟶ F' ⊛ G' ⊛ H') → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H')
          (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
            (Iso.hom (associator F G H)) (map f (map g h)))) :
      (F ⊠ G) ⊠ H ⟶
        Functor.prod (tensor C) (𝟭 C) ⋙
          tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _)

Tactic:
ext