-- 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 F
inst_13 : DayConvolution G (@convolution _ _ _ _ inst_2 inst_4 H F _)
inst_14 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 G H _) F
inst_15 : DayConvolution G F
inst_16 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 G F _) H
inst_17 : DayConvolution F H
inst_18 : DayConvolution G (@convolution _ _ _ _ inst_2 inst_4 F H _)
hγ :
  Eq (α :=
    (F ⊠ G) ⊠ H ⟶
      Functor.prod (tensor C) (𝟭 C) ⋙
        tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _)
    (ExternalProduct.extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_4 F G _) (unit F G) H ≫
      Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
        (CategoryStruct.comp (Y :=
          tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _)
          (unit (@convolution _ _ _ _ inst_2 inst_4 F G _) H)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _)
              (Iso.hom (associator F G H))
              (CategoryStruct.comp (Y :=
                @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 G H _) F _)
                (Iso.hom (braiding F (@convolution _ _ _ _ inst_2 inst_4 G H _))) (Iso.hom (associator G H F)))))))
    (ExternalProduct.extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_4 F G _) (unit F G) H ≫
      Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
        (CategoryStruct.comp (Y :=
          tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _)
          (unit (@convolution _ _ _ _ inst_2 inst_4 F G _) H)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 G F _) H _)
              (map (Iso.hom (braiding F G)) (𝟙 H))
              (CategoryStruct.comp (Y :=
                @convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 F H _) _)
                (Iso.hom (associator G F H)) (map (𝟙 G) (Iso.hom (braiding F H))))))))
⊢ Eq (α :=
    @convolution _ _ _ _ inst_2 inst_4 F G _ ⊠ H ⟶
      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _)
    (CategoryStruct.comp (Y :=
      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _)
      (unit (@convolution _ _ _ _ inst_2 inst_4 F G _) H)
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _)
          (Iso.hom (associator F G H))
          (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 G H _) F _)
            (Iso.hom (braiding F (@convolution _ _ _ _ inst_2 inst_4 G H _))) (Iso.hom (associator G H F))))))
    (CategoryStruct.comp (Y :=
      tensor C ⋙ @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _)
      (unit (@convolution _ _ _ _ inst_2 inst_4 F G _) H)
      (Functor.whiskerLeft (tensor C)
        (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 G F _) H _)
          (map (Iso.hom (braiding F G)) (𝟙 H))
          (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 F H _) _)
            (Iso.hom (associator G F H)) (map (𝟙 G) (Iso.hom (braiding F H)))))))

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

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