-- 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'
dsimp :
  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))
⊢ 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 ≫
      (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) ◁ᵢ
              Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U' ≪≫
            Iso.symm
                (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' ▷ᵢ
                  Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ≪≫
                α_ U' (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                  (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))).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) ◁ᵢ
              Y.snd (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X).β U ≪≫
            Iso.symm
                (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 ▷ᵢ
                  Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X) ≪≫
                α_ U (X.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))
                  (Y.fst (β := fun (X : C) ↦ HalfBraiding.{v₁, u₁} X))).hom ≫
      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:
dsimp only [Iso.trans_hom, whiskerLeftIso_hom, Iso.symm_hom, whiskerRightIso_hom]

Premises used: CategoryTheory.Iso.symm_hom, CategoryTheory.MonoidalCategory.whiskerLeftIso_hom, CategoryTheory.Iso.trans_hom, CategoryTheory.MonoidalCategory.whiskerRightIso_hom