-- 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)) ≫
        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 ≫ Iso.hom (α_ x z y) ≫ 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))))
    (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 F H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x z)) (G.obj y))
          (NatTrans.app (unit F H) (x, z) ▷ G.obj y :
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (H.obj z)) (G.obj y) ⟶
              tensorObj.{v₂, u₂} (𝒞 := inst_1)
                ((@convolution _ _ _ _ inst_2 inst_4 F H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x 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) x z)) (G.obj y))
              (NatTrans.app (Iso.hom (braiding F H)) (tensorObj.{v₁, u₁} (𝒞 := inst) x z) ▷ G.obj y)
              (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_4 H F _) G)
                  (tensorObj.{v₁, u₁} (𝒞 := inst) x z, y) ≫
                (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 H F _) G _).map
                  (Iso.hom (α_ x z y) ≫ x ◁ Iso.hom (β_ z y))) :
            tensorObj.{v₂, u₂} (𝒞 := inst_1)
                ((@convolution _ _ _ _ inst_2 inst_4 F H _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x z)) (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))))
⊢ 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))))))

Tactic:
simp only [comp_whiskerRight, ← Functor.map_comp, Category.assoc, Functor.comp_obj, tensor_obj,
    whiskerLeft_comp, whiskerLeft_comp_unit_app_assoc, NatTrans.naturality_assoc,
    NatTrans.naturality, associator_inv_unit_unit_assoc, externalProductBifunctor_obj_obj,
    unit_app_map_app_assoc, NatTrans.id_app, tensorHom_id]

Premises used: CategoryTheory.Functor.comp_obj, CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.externalProductBifunctor_obj_obj, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.MonoidalCategory.whiskerLeft_comp, CategoryTheory.MonoidalCategory.comp_whiskerRight, CategoryTheory.NatTrans.naturality, CategoryTheory.NatTrans.id_app, CategoryTheory.MonoidalCategory.DayConvolution.whiskerLeft_comp_unit_app_assoc, CategoryTheory.MonoidalCategory.DayConvolution.associator_inv_unit_unit_assoc, CategoryTheory.Functor.map_comp, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app_assoc, CategoryTheory.NatTrans.naturality_assoc, CategoryTheory.MonoidalCategory.tensorHom_id