-- 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'
x : (C × C) × C
a :
  Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (G.obj x.1.2)) (H.obj x.2) ⟶
      (@convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) x.2))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
        ((@convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2)) (H.obj x.2))
      (NatTrans.app (unit F G) x.1 ▷ H.obj x.2)
      (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          ((@convolution _ _ _ _ inst_2 inst_3 F' G' _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2)) (H'.obj x.2))
        (NatTrans.app (map f g) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) ⊗ₘ h.app x.2)
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F' G' _) H' _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) x.2))
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F' G' _) H')
            (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2, x.2))
          (NatTrans.app (Iso.hom (associator F' G' H'))
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) x.2)))))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
        ((@convolution _ _ _ _ inst_2 inst_3 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2)) (H.obj x.2))
      (NatTrans.app (unit F G) x.1 ▷ H.obj x.2)
      (CategoryStruct.comp (Y :=
        (@convolution _ _ _ _ inst_2 inst_3 (@convolution _ _ _ _ inst_2 inst_3 F G _) H _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) x.2))
        (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_3 F G _) H)
          (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2, x.2))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) x.2))
          (NatTrans.app (Iso.hom (associator F G H))
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) x.2))
          (NatTrans.app (map f (map g h))
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) x.2)))))
⊢ Eq.{v₂ + 1} (α :=
    ((F ⊠ G) ⊠ H).obj x ⟶
      (Functor.prod (tensor C) (𝟭 C) ⋙
            tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _).obj
        x)
    (NatTrans.app
      (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
            (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
            (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _) :
          (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H') → (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F' ⊛ G' ⊛ H')
        ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
            (a : (F ⊛ G) ⊛ H ⟶ F' ⊛ G' ⊛ H') → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ 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')))))
      x)
    (NatTrans.app
      (((@convolution _ _ _ _ inst_2 inst_3 F G _ ⊠ H).homEquivOfIsLeftKanExtension
            (extensionUnitLeft (@convolution _ _ _ _ inst_2 inst_3 F G _) (unit F G) H)
            (tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _) :
          (a : (F ⊛ G) ⊠ H ⟶ tensor C ⋙ F' ⊛ G' ⊛ H') → (F ⊠ G) ⊠ H ⟶ (tensor C).prod (𝟭 C) ⋙ tensor C ⋙ F' ⊛ G' ⊛ H')
        ((CorepresentableBy.homEquiv (corepresentableBy (@convolution _ _ _ _ inst_2 inst_3 F G _) H) :
            (a : (F ⊛ G) ⊛ H ⟶ F' ⊛ G' ⊛ H') → (F ⊛ G) ⊠ H ⟶ tensor C ⋙ 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)))))
      x)

Tactic:
simp only [externalProductBifunctor_obj_obj, Functor.comp_obj, Functor.prod_obj, tensor_obj,
    Functor.id_obj, Functor.homEquivOfIsLeftKanExtension_apply_app,
    externalProductBifunctor_map_app, Functor.leftUnitor_inv_app, whiskerLeft_id, Category.comp_id,
    corepresentableBy_homEquiv_apply_app, NatTrans.comp_app, unit_app_map_app_assoc]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.MonoidalCategory.externalProductBifunctor_obj_obj, CategoryTheory.MonoidalCategory.externalProductBifunctor_map_app, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.Functor.prod_obj, CategoryTheory.Category.comp_id, CategoryTheory.Functor.leftUnitor_inv_app, CategoryTheory.MonoidalCategory.whiskerLeft_id, CategoryTheory.MonoidalCategory.DayConvolution.corepresentableBy_homEquiv_apply_app, CategoryTheory.Functor.id_obj, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app_assoc, CategoryTheory.NatTrans.comp_app, CategoryTheory.Functor.homEquivOfIsLeftKanExtension_apply_app