-- Module: Mathlib.CategoryTheory.Monoidal.Center
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y : Center.{v₁, u₁} C
U U' : C
f : U ⟶ U'
_ :
  MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) ◁
        f ≫
      Iso.hom
          (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            U') ≫
        X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
            (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom ≫
          Iso.inv
              (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U'
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))) ≫
            (X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom ▷
                Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ≫
              Iso.hom
                (α_ U' (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                  (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))) =
    monoidalComp (Y :=
      MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U))
      (𝟙
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
          U))
      (monoidalComp (Y :=
        MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            U')
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
        (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁ f ≫
            (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom))
        ((X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom ▷
            Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ⊗≫
          𝟙
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U'
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
                (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))))))
_₁ :
  monoidalComp (Y :=
      MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U))
      (𝟙
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
          U))
      (monoidalComp (Y :=
        MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            U')
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
        (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁ f ≫
            (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom))
        ((X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom ▷
            Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ⊗≫
          𝟙
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U'
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
                (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))))) =
    monoidalComp (Y :=
      MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U))
      (𝟙
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
          U))
      (monoidalComp (Y :=
        MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U)
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
        (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
          (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U).hom)
        ((X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁ f ≫
              (X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom) ▷
            Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ⊗≫
          𝟙
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U'
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
                (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))))))
_₂ :
  monoidalComp (Y :=
      MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U))
      (𝟙
        (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
          U))
      (monoidalComp (Y :=
        MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U)
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
        (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
          (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U).hom)
        ((X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁ f ≫
              (X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom) ▷
            Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ⊗≫
          𝟙
            (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U'
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst)
                (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))))) =
    (Iso.hom
          (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            U) ≫
        X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
            (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U).hom ≫
          Iso.inv
              (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))) ≫
            (X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U).hom ▷
                Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ≫
              Iso.hom
                (α_ U (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                  (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))) ≫
      f ▷
        MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
⊢ MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) ◁
        f ≫
      Iso.hom
          (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            U') ≫
        X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
            (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom ≫
          Iso.inv
              (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U'
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))) ≫
            (X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom ▷
                Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ≫
              Iso.hom
                (α_ U' (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                  (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))) =
    (Iso.hom
          (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
            U) ≫
        X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
            (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U).hom ≫
          Iso.inv
              (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))) ≫
            (X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U).hom ▷
                Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ≫
              Iso.hom
                (α_ U (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                  (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))) ≫
      f ▷
        MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
          (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))

Tactic:
calc
          _ = 𝟙 _ ⊗≫
            (X.1 ◁ (Y.1 ◁ f ≫ (HalfBraiding.β Y.2 U').hom)) ⊗≫
              (HalfBraiding.β X.2 U').hom ▷ Y.1 ⊗≫ 𝟙 _ := by monoidal
          _ = 𝟙 _ ⊗≫
            X.1 ◁ (HalfBraiding.β Y.2 U).hom ⊗≫
              (X.1 ◁ f ≫ (HalfBraiding.β X.2 U').hom) ▷ Y.1 ⊗≫ 𝟙 _ := by
            rw [HalfBraiding.naturality]; monoidal
          _ = _ := by rw [HalfBraiding.naturality]; monoidal

Premises used: CategoryTheory.HalfBraiding.β, CategoryTheory.MonoidalCategoryStruct.whiskerLeft, CategoryTheory.CategoryStruct.id, Sigma.fst, CategoryTheory.CategoryStruct.comp, Sigma.snd, CategoryTheory.Iso.hom