-- 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)
F' G' H' : Functor.{v₁, v₂, u₁, u₂} C V
inst_10 : DayConvolution F' G'
inst_11 : DayConvolution G' H'
inst_12 : DayConvolution F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _)
inst_13 : DayConvolution (@convolution _ _ _ _ inst_2 inst_3 F' G' _) H'
f : F ⟶ F'
g : G ⟶ G'
h : H ⟶ H'
a :
  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
      (@convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _))
    ((CorepresentableBy.homEquiv (corepresentableBy₂' F G H) :
          (a : (F ⊛ G) ⊛ H ⟶ F' ⊛ G' ⊛ H') →
            ((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
              (F' ⊛ G' ⊛ H'))
        (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F' G' _) H' _)
          (map (map f g) h) (Iso.hom (associator F' G' H'))) :
      (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
        (@convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _))
    ((CorepresentableBy.homEquiv (corepresentableBy₂' F G H) :
          (a : (F ⊛ G) ⊛ H ⟶ F' ⊛ G' ⊛ H') →
            ((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
              (F' ⊛ G' ⊛ H'))
        (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
          (Iso.hom (associator F G H)) (map f (map g h))) :
      (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
        (@convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _))
⊢ Eq (α :=
    @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _ ⟶
      @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _)
    (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F' G' _) H' _)
      (map (map f g) h) (Iso.hom (associator F' G' H')))
    (CategoryStruct.comp (Y := @convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _)
      (Iso.hom (associator F G H)) (map f (map g h)))

Tactic:
apply (corepresentableBy₂' F G H) |>.homEquiv.injective

Premises used: CategoryTheory.Functor.CorepresentableBy.homEquiv, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy₂', Equiv.injective