-- 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)
      (NatTrans.app (map f g) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) ▷ H.obj x.2 ≫
        (@convolution _ _ _ _ inst_2 inst_3 F' G' _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.1 x.1.2) ◁ 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))))
    (Iso.hom (α_ (F.obj x.1.1) (G.obj x.1.2) (H.obj x.2)) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
          ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 x.2)))
        (F.obj x.1.1 ◁ NatTrans.app (unit G H) (x.1.2, x.2) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj x.1.2) (H.obj x.2)) ⟶
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
              ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 x.2)))
        (f.app x.1.1 ▷ (@convolution _ _ _ _ inst_2 inst_3 G H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 x.2) ≫
            CategoryStruct.comp (Y :=
              tensorObj.{v₂, u₂} (𝒞 := inst_1) (F'.obj x.1.1)
                ((@convolution _ _ _ _ inst_2 inst_3 G' H' _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 x.2)))
              (F'.obj x.1.1 ◁ NatTrans.app (map g h) (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 x.2))
              (NatTrans.app (unit F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _))
                  (x.1.1, tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 x.2) ≫
                (@convolution _ _ _ _ inst_2 inst_3 F' (@convolution _ _ _ _ inst_2 inst_3 G' H' _) _).map
                  (Iso.inv (α_ x.1.1 x.1.2 x.2))) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
              ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 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)))
⊢ 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)))))
    (Iso.hom (α_ (F.obj x.1.1) (G.obj x.1.2) (H.obj x.2)) ≫
      CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
          ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (obj.{v₁, v₁, u₁, u₁} (tensor C) (x.1.2, x.2))))
        (F.obj x.1.1 ◁ NatTrans.app (unit G H) (x.1.2, x.2) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj x.1.2) (H.obj x.2)) ⟶
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
              ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (obj.{v₁, v₁, u₁, u₁} (tensor C) (x.1.2, x.2))))
        (NatTrans.app (unit F (@convolution _ _ _ _ inst_2 inst_3 G H _))
              (x.1.1, tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 x.2) ≫
            (@convolution _ _ _ _ inst_2 inst_3 F (@convolution _ _ _ _ inst_2 inst_3 G H _) _).map
                (Iso.inv (α_ 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) :
          tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x.1.1)
              ((@convolution _ _ _ _ inst_2 inst_3 G H _).obj (obj.{v₁, v₁, u₁, u₁} (tensor C) (x.1.2, 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)))

Tactic:
simp only [tensorHom_def, Category.assoc, externalProductBifunctor_obj_obj, tensor_obj,
    NatTrans.naturality, unit_app_map_app_assoc]

Premises used: CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.externalProductBifunctor_obj_obj, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.MonoidalCategory.tensorHom_def, CategoryTheory.NatTrans.naturality, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app_assoc