-- 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 F
inst_13 : DayConvolution G (@convolution _ _ _ _ inst_2 inst_4 H F _)
inst_14 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 G H _) F
inst_15 : DayConvolution G F
inst_16 : DayConvolution (@convolution _ _ _ _ inst_2 inst_4 G F _) H
inst_17 : DayConvolution F H
inst_18 : DayConvolution G (@convolution _ _ _ _ inst_2 inst_4 F H _)
z x y : C
hγ :
  Eq.{v₂ + 1} (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
      (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (Iso.hom (β_ (F.obj x) (G.obj y)) ▷ H.obj z ≫
      Iso.hom (α_ (G.obj y) (F.obj x) (H.obj z)) ≫
        G.obj y ◁ Iso.hom (β_ (F.obj x) (H.obj z)) ≫
          CategoryStruct.comp (Y :=
            tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y)
              ((@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) :
              tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (H.obj z) (F.obj x)) ⟶
                tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y)
                  ((@convolution _ _ _ _ inst_2 inst_4 H F _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) z x)))
            (NatTrans.app (unit G (@convolution _ _ _ _ inst_2 inst_4 H F _)) (y, tensorObj.{v₁, u₁} (𝒞 := inst) z x) ≫
                (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _).map
                    (y ◁ Iso.hom (β_ z x)) ≫
                  (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _).map
                      (Iso.inv (α_ y x z)) ≫
                    (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _).map
                      (Iso.hom (β_ y x) ▷ z) :
              tensorObj.{v₂, u₂} (𝒞 := inst_1) (G.obj y)
                  ((@convolution _ _ _ _ inst_2 inst_4 H F _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) z x)) ⟶
                (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _).obj
                  (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z)))
    (CategoryStruct.comp (Y :=
      tensorObj.{v₂, u₂} (𝒞 := inst_1)
        ((@convolution _ _ _ _ inst_2 inst_4 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) (H.obj z))
      (NatTrans.app (unit F G) (x, y) ▷ H.obj z)
      (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          ((@convolution _ _ _ _ inst_2 inst_4 G F _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) (H.obj z))
        (NatTrans.app (Iso.hom (braiding F G)) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) ▷ H.obj z)
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 G F _) H _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_4 G F _) H) (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z))
          (CategoryStruct.comp (Y :=
            (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 F H _) _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
            (NatTrans.app (Iso.hom (associator G F H))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
            (NatTrans.app (map (𝟙 G) (Iso.hom (braiding F H)))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))))))
⊢ Eq (α :=
    tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
      (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _).obj
        (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          ((@convolution _ _ _ _ inst_2 inst_4 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) (H.obj z))
        (NatTrans.app (unit F G) (x, y) ▷ H.obj z ≫
          (@convolution _ _ _ _ inst_2 inst_4 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y) ◁ 𝟙 (H.obj z))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_4 F G _) H) (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z))
          (CategoryStruct.comp (Y :=
            (@convolution _ _ _ _ inst_2 inst_4 F (@convolution _ _ _ _ inst_2 inst_4 G H _) _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
            (NatTrans.app (Iso.hom (associator F G H))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
            (CategoryStruct.comp (Y :=
              (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 G H _) F _).obj
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
              (NatTrans.app (Iso.hom (braiding F (@convolution _ _ _ _ inst_2 inst_4 G H _)))
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
              (NatTrans.app (Iso.hom (associator G H F))
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))))) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
        (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
    (CategoryStruct.comp (Y :=
        tensorObj.{v₂, u₂} (𝒞 := inst_1)
          ((@convolution _ _ _ _ inst_2 inst_4 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y)) (H.obj z))
        (NatTrans.app (unit F G) (x, y) ▷ H.obj z ≫
          (@convolution _ _ _ _ inst_2 inst_4 F G _).obj (tensorObj.{v₁, u₁} (𝒞 := inst) x y) ◁ 𝟙 (H.obj z))
        (CategoryStruct.comp (Y :=
          (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 F G _) H _).obj
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
          (NatTrans.app (unit (@convolution _ _ _ _ inst_2 inst_4 F G _) H) (tensorObj.{v₁, u₁} (𝒞 := inst) x y, z))
          (CategoryStruct.comp (Y :=
            (@convolution _ _ _ _ inst_2 inst_4 (@convolution _ _ _ _ inst_2 inst_4 G F _) H _).obj
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
            (NatTrans.app (map (Iso.hom (braiding F G)) (𝟙 H))
              (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
            (CategoryStruct.comp (Y :=
              (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 F H _) _).obj
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
              (NatTrans.app (Iso.hom (associator G F H))
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))
              (NatTrans.app (map (𝟙 G) (Iso.hom (braiding F H)))
                (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))))) :
      tensorObj.{v₂, u₂} (𝒞 := inst_1) (tensorObj.{v₂, u₂} (𝒞 := inst_1) (F.obj x) (G.obj y)) (H.obj z) ⟶
        (@convolution _ _ _ _ inst_2 inst_4 G (@convolution _ _ _ _ inst_2 inst_4 H F _) _).obj
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) x y) z))

Tactic:
simp only [whiskerLeft_id, Category.comp_id, associator_hom_unit_unit_assoc,
    externalProductBifunctor_obj_obj, tensor_obj, NatTrans.naturality_assoc,
    NatTrans.naturality, unit_app_braiding_hom_app_assoc,
    BraidedCategory.braiding_tensor_left_hom, Functor.map_comp, Category.assoc,
    Iso.map_hom_inv_id, BraidedCategory.braiding_naturality_right_assoc,
    BraidedCategory.braiding_tensor_right_hom, Iso.map_inv_hom_id_assoc,
    Iso.inv_hom_id_assoc, Iso.hom_inv_id_assoc, unit_app_map_app_assoc,
    NatTrans.id_app, tensorHom_id]

Premises used: CategoryTheory.BraidedCategory.braiding_tensor_right_hom, CategoryTheory.Category.assoc, CategoryTheory.MonoidalCategory.externalProductBifunctor_obj_obj, CategoryTheory.MonoidalCategory.tensor_obj, CategoryTheory.BraidedCategory.braiding_tensor_left_hom, CategoryTheory.Iso.map_hom_inv_id, CategoryTheory.Category.comp_id, CategoryTheory.Iso.inv_hom_id_assoc, CategoryTheory.Iso.map_inv_hom_id_assoc, CategoryTheory.MonoidalCategory.whiskerLeft_id, CategoryTheory.NatTrans.naturality, CategoryTheory.MonoidalCategory.DayConvolution.associator_hom_unit_unit_assoc, CategoryTheory.Iso.hom_inv_id_assoc, CategoryTheory.NatTrans.id_app, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_braiding_hom_app_assoc, CategoryTheory.Functor.map_comp, CategoryTheory.MonoidalCategory.DayConvolution.unit_app_map_app_assoc, CategoryTheory.NatTrans.naturality_assoc, CategoryTheory.BraidedCategory.braiding_naturality_right_assoc, CategoryTheory.MonoidalCategory.tensorHom_id