-- 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
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_7 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (tensor C) (𝟭 C)) d)
      (tensorRight v)
H K : Functor.{v₁, v₂, u₁, u₂} C V
inst_8 : DayConvolution G H
inst_9 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F G _) H
inst_10 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 G H _)
inst_11 : DayConvolution H K
inst_12 : DayConvolution G (@convolution _ _ _ _ inst_2 inst_3 H K _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 G H _) K
inst_14 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K
inst_15 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F G _) (@convolution _ _ _ _ inst_2 inst_3 H K _)
inst_16 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) K
inst_17 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _)
inst_18 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 G H _) K _)
this :
  ((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H) ⊠ K).IsLeftKanExtension
    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H)
      (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H) K)
hγ :
  Eq (α :=
    ((F ⊠ G) ⊠ H) ⊠ K ⟶
      Functor.prod (Functor.prod (tensor C) (𝟭 C)) (𝟭 C) ⋙
        Functor.prod (tensor C) (𝟭 C) ⋙
          tensor C ⋙
            @convolution _ _ _ _ inst_2 inst_3 F
              (@convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _) _)
    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H)
        (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H) K ≫
      Functor.whiskerLeft (Functor.prod (Functor.prod (tensor C) (𝟭 C)) (𝟭 C))
        (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _)
            (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H) K ≫
          Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
            (CategoryStruct.comp (Y :=
              tensor C ⋙
                @convolution _ _ _ _ inst_2 inst_3
                  (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K _)
              (unit (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K)
              (Functor.whiskerLeft (tensor C)
                (CategoryStruct.comp (Y :=
                  @convolution _ _ _ _ inst_2 inst_3
                    (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) K _)
                  (map (Iso.hom (associator F G H)) (𝟙 K))
                  (CategoryStruct.comp (Y :=
                    @convolution _ _ _ _ inst_2 inst_3 F
                      (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 G H _) K _) _)
                    (Iso.hom (associator F (@convolution _ _ _ _ inst_2 inst_3 G H _) K))
                    (map (𝟙 F) (Iso.hom (associator G H K)))))))))
    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H)
        (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H) K ≫
      Functor.whiskerLeft (Functor.prod (Functor.prod (tensor C) (𝟭 C)) (𝟭 C))
        (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _)
            (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H) K ≫
          Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
            (CategoryStruct.comp (Y :=
              tensor C ⋙
                @convolution _ _ _ _ inst_2 inst_3
                  (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K _)
              (unit (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K)
              (Functor.whiskerLeft (tensor C)
                (CategoryStruct.comp (Y :=
                  @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _)
                    (@convolution _ _ _ _ inst_2 inst_3 H K _) _)
                  (Iso.hom (associator (@convolution _ _ _ _ inst_2 inst_3 F G _) H K))
                  (Iso.hom (associator F G (@convolution _ _ _ _ inst_2 inst_3 H K _))))))))
⊢ Eq (α :=
    (@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H) ⊠ K ⟶
      Functor.prod (tensor C) (𝟭 C) ⋙
        tensor C ⋙
          @convolution _ _ _ _ inst_2 inst_3 F
            (@convolution _ _ _ _ inst_2 inst_3 G (@convolution _ _ _ _ inst_2 inst_3 H K _) _) _)
    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _)
        (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H) K ≫
      Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
        (CategoryStruct.comp (Y :=
          tensor C ⋙
            @convolution _ _ _ _ inst_2 inst_3
              (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K _)
          (unit (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_3
                (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _) K _)
              (map (Iso.hom (associator F G H)) (𝟙 K))
              (CategoryStruct.comp (Y :=
                @convolution _ _ _ _ inst_2 inst_3 F
                  (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 G H _) K _) _)
                (Iso.hom (associator F (@convolution _ _ _ _ inst_2 inst_3 G H _) K))
                (map (𝟙 F) (Iso.hom (associator G H K))))))))
    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _)
        (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H) K ≫
      Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
        (CategoryStruct.comp (Y :=
          tensor C ⋙
            @convolution _ _ _ _ inst_2 inst_3
              (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K _)
          (unit (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _) K)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _)
                (@convolution _ _ _ _ inst_2 inst_3 H K _) _)
              (Iso.hom (associator (@convolution _ _ _ _ inst_2 inst_3 F G _) H K))
              (Iso.hom (associator F G (@convolution _ _ _ _ inst_2 inst_3 H K _)))))))

Tactic:
apply Functor.hom_ext_of_isLeftKanExtension (α := extensionUnitLeft ((F ⊛ G) ⊠ H)
      (extensionUnitLeft _ (unit F G) H) K)

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