-- 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) ≫
                  (@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)))))))
⊢ Eq (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)))
    (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)))
        (𝟙 (F.obj x) ▷ tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z) ≫ F.obj x ◁ NatTrans.app (unit G H) (y, z))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
          (NatTrans.app (unit F (@convolution _ _ _ _ inst_2 inst_4 G H _)) (x, tensorObj.{v₁, u₁} (𝒞 := inst) y z))
          (CategoryStruct.comp (Y :=
            (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
            (NatTrans.app (Iso.inv (associator F G H))
              (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
            (CategoryStruct.comp (Y :=
              (@convolution _ _ _ _ inst_2 inst_4 H (@convolution _ _ _ _ inst_2 inst_4 F G _) _).obj
                (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
              (NatTrans.app (Iso.hom (braiding (@convolution _ _ _ _ inst_2 inst_4 F G _) H))
                (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
              (NatTrans.app (Iso.inv (associator H F G))
                (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))))) :
      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)))
    (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)))
        (𝟙 (F.obj x) ▷ tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (H.obj z) ≫ F.obj x ◁ NatTrans.app (unit G H) (y, z))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) x (tensorObj.{v₁, u₁} (𝒞 := inst) y z)))
          (NatTrans.app (unit F (@convolution _ _ _ _ inst_2 inst_4 G H _)) (x, 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 (map (𝟙 F) (Iso.hom (braiding G H)))
              (tensorObj.{v₁, u₁} (𝒞 := inst) 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)))))) :
      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)))

Tactic:
simp only [whiskerRight_tensor, id_whiskerRight, Category.id_comp,
    Iso.inv_hom_id, associator_inv_unit_unit_assoc,
    externalProductBifunctor_obj_obj, tensor_obj, NatTrans.naturality_assoc,
    NatTrans.naturality, unit_app_braiding_hom_app_assoc,
    BraidedCategory.braiding_tensor_right_hom, Functor.map_comp, Category.assoc,
    Iso.map_inv_hom_id, Category.comp_id,
    BraidedCategory.braiding_naturality_left_assoc,
    BraidedCategory.braiding_tensor_left_hom, Iso.map_hom_inv_id_assoc,
    Iso.hom_inv_id_assoc, Iso.inv_hom_id_assoc, unit_app_map_app_assoc,
    NatTrans.id_app, id_tensorHom]

Premises used: CategoryTheory.BraidedCategory.braiding_tensor_right_hom, CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.externalProductBifunctor_obj_obj, CategoryTheory.Iso.inv_hom_id, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.MonoidalCategory.whiskerRight_tensor, CategoryTheory.MonoidalCategory.id_tensorHom, CategoryTheory.BraidedCategory.braiding_tensor_left_hom, CategoryTheory.Iso.map_hom_inv_id_assoc, CategoryTheory.Category.comp_id, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.MonoidalCategory.id_whiskerRight, CategoryTheory.Iso.map_inv_hom_id, CategoryTheory.NatTrans.naturality, CategoryTheory.Iso.hom_inv_id_assoc, CategoryTheory.BraidedCategory.braiding_naturality_left_assoc, CategoryTheory.NatTrans.id_app, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_braiding_hom_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.Category.id_comp