-- 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
inst_4 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_5 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
inst_6 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C),
    Limits.PreservesColimitsOfShape.{0, v₁, v₂, v₂, u₂, u₂}
      (CostructuredArrow (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))) d) (tensorRight v)
inst_8 :
  ∀ (v : V) (d : C × C),
    Limits.PreservesColimitsOfShape
      (CostructuredArrow.{v₁, v₁, u₁, u₁} (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst))))
        d)
      (tensorRight v)
F G U : Functor.{v₁, v₂, u₁, u₂} C V
inst_9 : DayConvolutionUnit U
inst_10 : DayConvolution F U
inst_11 : DayConvolution U G
inst_12 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_3 U G _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F U _) G
inst_14 : DayConvolution F G
this :
  ((F ⊠ U) ⊠ G).IsLeftKanExtension (L :=
    Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C)) (F :=
    (F ⊠ Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) ⊠ G)
    (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G)
hγ :
  Eq (α :=
    (F ⊠ Functor.fromPUnit (tensorUnit.{v₂, u₂} V (𝒞 := inst_1))) ⊠ G ⟶
      Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C) ⋙
        Functor.prod (tensor C) (𝟭 C) ⋙ tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F G _)
    (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G ≫
      Functor.whiskerLeft
        (Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C))
        (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F U _) (unit F U) G ≫
          Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
            (CategoryStruct.comp (Y :=
              tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
              (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
              (Functor.whiskerLeft (tensor C)
                (CategoryStruct.comp (Y :=
                  @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _)
                  (Iso.hom (associator F U G)) (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G))))))))
    (extensionUnitLeft (F ⊠ U) (extensionUnitRight U (φ U) F) G ≫
      Functor.whiskerLeft
        (Functor.prod (Functor.prod (𝟭 C) (Functor.fromPUnit (tensorUnit.{v₁, u₁} C (𝒞 := inst)))) (𝟭 C))
        (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F U _) (unit F U) G ≫
          Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
            (CategoryStruct.comp (Y :=
              tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
              (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
              (Functor.whiskerLeft (tensor C) (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G))))))
⊢ Eq (α := (F ⊠ U) ⊠ G ⟶ Functor.prod (tensor C) (𝟭 C) ⋙ tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F G _)
    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F U _) (unit F U) G ≫
      Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
        (CategoryStruct.comp (Y :=
          tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
          (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
          (Functor.whiskerLeft (tensor C)
            (CategoryStruct.comp (Y :=
              @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 U G _) _)
              (Iso.hom (associator F U G)) (map (𝟙 F) (Iso.hom (DayConvolutionUnit.leftUnitor U G)))))))
    (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F U _) (unit F U) G ≫
      Functor.whiskerLeft (Functor.prod (tensor C) (𝟭 C))
        (CategoryStruct.comp (Y :=
          tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F U _) G _)
          (unit (@convolution _ _ _ _ inst_2 inst_3 F U _) G)
          (Functor.whiskerLeft (tensor C) (map (Iso.hom (DayConvolutionUnit.rightUnitor U F)) (𝟙 G)))))

Tactic:
apply Functor.hom_ext_of_isLeftKanExtension
    (α := extensionUnitLeft (F ⊠ U) (extensionUnitRight U (DayConvolutionUnit.φ U) F) G)

Premises used: CategoryTheory.Functor.hom_ext_of_isLeftKanExtension, CategoryTheory.MonoidalCategory.ExternalProduct.extensionUnitRight, CategoryTheory.MonoidalCategory.DayConvolutionUnit.φ, CategoryTheory.MonoidalCategory.externalProduct, CategoryTheory.MonoidalCategory.ExternalProduct.extensionUnitLeft