-- 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))
    ((f.app x.1.1 ⊗ₘ g.app x.1.2) ▷ H.obj x.2 ≫
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F'.obj x.1.1) (G'.obj x.1.2) ◁ h.app x.2 ≫
        Iso.hom (α_ (F'.obj x.1.1) ((G', H').1.obj (x.1.2, x.2).1) ((G', H').2.obj (x.1.2, x.2).2)) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (F'.obj x.1.1)
              ((tensor C ⋙ @convolution _ _ _ _ inst_2 inst_3 G' H' _).obj (x.1.2, x.2)))
            (F'.obj x.1.1 ◁ NatTrans.app (unit G' H') (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, tensorObj.{v₁, u₁} (𝒞 := inst) x.1.2 x.2).1 x.1.2 x.2))))
    (Iso.hom (α_ (F.obj x.1.1) (G.obj x.1.2) (H.obj x.2)) ≫
      f.app x.1.1 ▷ tensorObj.{v₂, u₂} (𝒞 := inst_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 ◁ ((g.app x.1.2 ⊗ₘ h.app x.2) ≫ 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)))
          (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))
    ((f.app x.1.1 ⊗ₘ g.app x.1.2) ▷ H.obj x.2 ≫
      NatTrans.app (unit F' G') (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)))

Tactic:
rw [← whisker_exchange_assoc, associator_hom_unit_unit, whisker_exchange_assoc,
    ← MonoidalCategory.whiskerLeft_comp_assoc, unit_app_map_app]

Premises used: CategoryTheory.MonoidalCategory.whisker_exchange_assoc, CategoryTheory.MonoidalCategory.whiskerLeft_comp_assoc, CategoryTheory.MonoidalCategory.DayConvolution.associator_hom_unit_unit, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app