-- Module: Mathlib.CategoryTheory.Monoidal.DayConvolution.Braided
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 : BraidedCategory.{v₁, u₁} C
inst_4 : MonoidalCategory.{v₂, u₂} V
inst_5 : BraidedCategory.{v₂, u₂} V
F G : Functor.{v₁, v₂, u₁, u₂} C V
inst_6 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorLeft v)
inst_7 :
  ∀ (v : V) (d : C), Limits.PreservesColimitsOfShape (CostructuredArrow.{v₁, v₁, u₁, u₁} (tensor C) d) (tensorRight v)
H : Functor.{v₁, v₂, u₁, u₂} C V
inst_8 : DayConvolution F G
inst_9 : DayConvolution G H
inst_10 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_4 G H _)
inst_11 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 F G _) H
inst_12 : DayConvolution H (@convolution _ _ _ _ inst_2 inst_4 F G _)
inst_13 : DayConvolution H F
inst_14 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 H F _) G
inst_15 : DayConvolution H G
inst_16 : DayConvolution F (@convolution _ _ _ _ inst_2 inst_4 H G _)
inst_17 : DayConvolution F H
inst_18 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 F H _) G
x y z : C
hγ :
  Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z)) ⟶
      (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
    (F.obj x ◁ Iso.hom (β_ (G.obj y) (H.obj z)) ≫
      Iso.inv (α_ (F.obj x) (H.obj z) (G.obj y)) ≫
        CategoryStruct.comp (Y :=
          tensorObj.{v₂, u₂} (𝒞 := inst_1)
            ((@convolution _ _ _ _ inst_2 inst_4 H F _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) z x)) (G.obj y))
          ((Iso.hom (β_ (F.obj x) (H.obj z)) ≫ NatTrans.app (unit H F) (z, x)) ▷ G.obj y)
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_4 H F _) G) (tensorObj.{v₁, u₁} (𝒞 := inst) z x, y) ≫
            (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).map
                (Iso.hom (β_ z x) ▷ y) ≫
              (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).map
                  (Iso.hom (α_ x z y)) ≫
                (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).map
                  (x ◁ Iso.hom (β_ z y))))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x)
        ((@convolution _ _ _ _ inst_2 inst_4 H G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
      (F.obj x ◁
        (Iso.hom (β_ (G.obj y) (H.obj z)) ≫
          NatTrans.app (unit H G) (z, y) ≫ (@convolution _ _ _ _ inst_2 inst_4 H G _).map (Iso.hom (β_ z y))))
      (CategoryStruct.comp (Y :=
        (@convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 H G _) _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
        (NatTrans.app (unit F (@convolution _ _ _ _ inst_2 inst_4 H G _)) (x, tensorObj.{v₁, u₁} (𝒞 := inst) y z))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F H _) G _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
          (NatTrans.app (Iso.inv (associator F H G))
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
          (NatTrans.app (map (Iso.hom (braiding F H)) (𝟙 G))
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z))))))
⊢ Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z)) ⟶
      (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
    (F.obj x ◁ Iso.hom (β_ (G.obj y) (H.obj z)) ≫
      Iso.inv (α_ (F.obj x) (H.obj z) (G.obj y)) ≫
        Iso.hom (β_ (F.obj x) (H.obj z)) ▷ G.obj y ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v₂, u₂} (𝒞 := inst_1)
              ((@convolution _ _ _ _ inst_2 inst_4 H F _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) z x)) (G.obj y))
            (NatTrans.app (unit H F) (z, x) ▷ G.obj y :
              tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (H.obj z) (F.obj x)) (G.obj y) ⟶
                tensorObj.{v₂, u₂} (𝒞 := inst_1)
                  ((@convolution _ _ _ _ inst_2 inst_4 H F _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) z x)) (G.obj y))
            (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_4 H F _) G) (tensorObj.{v₁, u₁} (𝒞 := inst) z x, y) ≫
                (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).map
                    (Iso.hom (β_ z x) ▷ y) ≫
                  (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).map
                      (Iso.hom (α_ x z y)) ≫
                    (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).map
                      (x ◁ Iso.hom (β_ z y)) :
              tensorObj.{v₂, u₂} (𝒞 := inst_1)
                  ((@convolution _ _ _ _ inst_2 inst_4 H F _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) z x)) (G.obj y) ⟶
                (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).obj
                  (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z))))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x)
        ((@convolution _ _ _ _ inst_2 inst_4 G H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
      (F.obj x ◁ NatTrans.app (unit G H) (y, z))
      (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x)
          ((@convolution _ _ _ _ inst_2 inst_4 H G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
        (F.obj x ◁ NatTrans.app (Iso.hom (braiding G H)) (tensorObj.{v₁, u₁} (𝒞 := inst) y z))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 H G _) _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
          (NatTrans.app (unit F (@convolution _ _ _ _ inst_2 inst_4 H G _)) (x, tensorObj.{v₁, u₁} (𝒞 := inst) y z))
          (CategoryStruct.comp (Y :=
            (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F H _) G _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
            (NatTrans.app (Iso.inv (associator F H G))
              (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
            (NatTrans.app (map (Iso.hom (braiding F H)) (𝟙 G))
              (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))))))

Tactic:
simp only [← comp_whiskerRight_assoc, ← whiskerLeft_comp_assoc,
    unit_app_braiding_hom_app]

Premises used: CategoryTheory.MonoidalCategory.DayConvolution.unit_app_braiding_hom_app, CategoryTheory.MonoidalCategory.comp_whiskerRight_assoc, CategoryTheory.MonoidalCategory.whiskerLeft_comp_assoc