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

Tactic:
simp [← Functor.map_comp]

Premises used: CategoryTheory.Functor.map_comp