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

Tactic:
simp [← Functor.map_comp]

Premises used: CategoryTheory.Functor.map_comp