-- 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)
⊢ ∀ {Y Y' : Functor.{v₁, v₂, u₁, u₂} C V} (g : Y ⟶ Y')
    (f : @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _ ⟶ Y),
    Eq (α :=
      (obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
            obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
              obj coyoneda (Opposite.op ((F ⊠ G) ⊠ H))).obj
        Y')
      ((Equiv.trans (CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H))
              ((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
                (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
                (obj (obj (whiskeringLeft (C × C) C V) (tensor C)) Y')) :
            (a : (F ⊛ G) ⊛ H ⟶ Y') →
              ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                    (whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                      coyoneda.obj (Opposite.op ((F ⊠ G) ⊠ H))).obj
                Y')
          (f ≫ g) :
        (obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
              obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
                obj coyoneda (Opposite.op ((F ⊠ G) ⊠ H))).obj
          Y')
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
                    obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
                      obj coyoneda (Opposite.op ((F ⊠ G) ⊠ H))).map
                g) :
            (a :
                (fun (X : Type (max u₁ v₂)) ↦ X)
                  (((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                        (whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                          coyoneda.obj (Opposite.op ((F ⊠ G) ⊠ H))).obj
                    Y)) →
              (fun (X : Type (max u₁ v₂)) ↦ X)
                (((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                      (whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                        coyoneda.obj (Opposite.op ((F ⊠ G) ⊠ H))).obj
                  Y'))
          ((Equiv.trans (CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H))
                ((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
                  (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
                  (obj (obj (whiskeringLeft (C × C) C V) (tensor C)) Y)) :
              (a : (F ⊛ G) ⊛ H ⟶ Y) →
                ((whiskeringLeft (C × C) C V).obj (tensor C) ⋙
                      (whiskeringLeft ((C × C) × C) (C × C) V).obj ((tensor C).prod (𝟭 C)) ⋙
                        coyoneda.obj (Opposite.op ((F ⊠ G) ⊠ H))).obj
                  Y)
            f) :
        (fun (X : Type (max u₁ v₂)) ↦ X)
          ((obj (whiskeringLeft (C × C) C V) (tensor C) ⋙
                obj (whiskeringLeft ((C × C) × C) (C × C) V) (Functor.prod (tensor C) (𝟭 C)) ⋙
                  obj coyoneda (Opposite.op ((F ⊠ G) ⊠ H))).obj
            Y'))

Tactic:
aesop