-- 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
⊢ Iso.hom
        (α_ (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) U U')) ≫
      X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ◁
          (Iso.inv (α_ (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U U') ≫
            (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U).hom ▷ U' ≫
              Iso.hom (α_ U (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U') ≫
                U ◁ (Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom ≫
                  Iso.inv (α_ U U' (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))) ≫
        Iso.inv
            (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
              (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U U')
              (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))) ≫
          (Iso.inv (α_ (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U U') ≫
                (X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U).hom ▷ U' ≫
                  Iso.hom (α_ U (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U') ≫
                    U ◁ (X.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U').hom ≫
                      Iso.inv (α_ U U' (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))) ▷
              Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ≫
            Iso.hom
              (α_ (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U 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)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)) U)
          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)))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U 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)
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (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 ▷ U')
        (monoidalComp (Y :=
          MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U
            (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))))
          (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                U ◁
              (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 ▷
              MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) U'
                (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X)))
          (U ◁
              (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)
                (MonoidalCategoryStruct.tensorObj.{v₁, u₁} (𝒞 := inst) 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)))))))

Tactic:
monoidal