-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided
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 : BraidedCategory.{v₁, u₁} C
inst_4 : MonoidalCategory.{v₂, u₂} V
inst_5 : BraidedCategory.{v₂, u₂} V
F G : Functor.{v₁, v₂, u₁, u₂} C V
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
H : Functor.{v₁, v₂, u₁, u₂} C V
inst_8 : DayConvolution F G
inst_9 : DayConvolution G H
inst_10 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_4 G H _)
inst_11 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 F G _) H
inst_12 : DayConvolution H (@convolution _ _ _ _ inst_2 inst_4 F G _)
inst_13 : DayConvolution H F
inst_14 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 H F _) G
inst_15 : DayConvolution H G
inst_16 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_4 H G _)
inst_17 : DayConvolution F H
inst_18 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 F H _) G
hγ :
  Eq (α :=
    F ⊠ G ⊠ H ⟶
      Functor.prod (𝟭 C) (tensor C) ⋙
        tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _)
    (ExternalProduct.extensionUnitRight (@convolution _ _ _ _ inst_2 inst_4 G H _) (unit G H) F ≫
      Functor.whiskerLeft (Functor.prod (𝟭 C) (tensor C))
        (CategoryStruct.comp (Y :=
          tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _)
          (unit F (@convolution _ _ _ _ inst_2 inst_4 G H _))
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _)
              (Iso.inv (associator F G H))
              (CategoryStruct.comp (Y :=
                @convolution _ _ _ _ inst_2 inst_4 H (@convolution _ _ _ _ inst_2 inst_4 F G _) _)
                (Iso.hom (braiding (@convolution _ _ _ _ inst_2 inst_4 F G _) H)) (Iso.inv (associator H F G)))))))
    (ExternalProduct.extensionUnitRight (@convolution _ _ _ _ inst_2 inst_4 G H _) (unit G H) F ≫
      Functor.whiskerLeft (Functor.prod (𝟭 C) (tensor C))
        (CategoryStruct.comp (Y :=
          tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _)
          (unit F (@convolution _ _ _ _ inst_2 inst_4 G H _))
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 H G _) _)
              (map (𝟙 F) (Iso.hom (braiding G H)))
              (CategoryStruct.comp (Y :=
                @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F H _) G _)
                (Iso.inv (associator F H G)) (map (Iso.hom (braiding F H)) (𝟙 G)))))))
⊢ Eq (α :=
    F ⊠ @convolution _ _ _ _ inst_2 inst_4 G H _ ⟶
      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _)
    (CategoryStruct.comp (Y :=
      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _)
      (unit F (@convolution _ _ _ _ inst_2 inst_4 G H _))
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _)
          (Iso.inv (associator F G H))
          (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_4 H (@convolution _ _ _ _ inst_2 inst_4 F G _) _)
            (Iso.hom (braiding (@convolution _ _ _ _ inst_2 inst_4 F G _) H)) (Iso.inv (associator H F G))))))
    (CategoryStruct.comp (Y :=
      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _)
      (unit F (@convolution _ _ _ _ inst_2 inst_4 G H _))
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 H G _) _)
          (map (𝟙 F) (Iso.hom (braiding G H)))
          (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F H _) G _)
            (Iso.inv (associator F H G)) (map (Iso.hom (braiding F H)) (𝟙 G))))))

Tactic:
apply Functor.hom_ext_of_isLeftKanExtension (F ⊠ (G ⊛ H))
    (ExternalProduct.extensionUnitRight (G ⊛ H) (unit G H) F)

Premises used: CategoryTheory.MonoidalCategory.DayConvolution.convolution, CategoryTheory.Functor.hom_ext_of_isLeftKanExtension, CategoryTheory.MonoidalCategory.ExternalProduct.extensionUnitRight, CategoryTheory.MonoidalCategory.externalProduct, CategoryTheory.MonoidalCategory.DayConvolution.unit