-- Module: Mathlib.CategoryTheory.Monoidal.Rigid.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
inst_1 : MonoidalCategory.{v₁, u₁} C
X Y Z : C
inst_2 : HasLeftDual.{v₁, u₁} X
inst_3 : HasLeftDual.{v₁, u₁} Y
inst_4 : HasLeftDual.{v₁, u₁} Z
f : X ⟶ Y
g : Y ⟶ Z
rw :
  η_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫
      CategoryStruct.comp (Y :=
            tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _))
            (Iso.inv (λ_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _)))
            (η_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ≫
              (@leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ f) ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ≫
                Iso.hom (α_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) Y (@leftDual.{v₁, u₁} _ inst inst_1 Y _)) ≫
                  @leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫
                    Iso.hom (ρ_ (@leftDual.{v₁, u₁} _ inst inst_1 X _))) ▷
          Y ≫
        @leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ g =
    monoidalComp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (𝟙 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (monoidalComp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X (@leftDual.{v₁, u₁} _ inst inst_1 Y _)) Y))
        (tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ η_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫
          η_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X ▷
            tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y)
        (monoidalComp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _)
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Y (@leftDual.{v₁, u₁} _ inst inst_1 Y _))
              Y))
          (@leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ f ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ▷ Y)
          (@leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ▷ Y ⊗≫
            @leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ g)))
⊢ η_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫
      (CategoryStruct.comp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (@leftDual.{v₁, u₁} _ inst inst_1 Y _))
          (Iso.inv (λ_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _)))
          (η_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ≫
            (@leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ f) ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ≫
              Iso.hom (α_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) Y (@leftDual.{v₁, u₁} _ inst inst_1 Y _)) ≫
                @leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫
                  Iso.hom (ρ_ (@leftDual.{v₁, u₁} _ inst inst_1 X _))) ⊗ₘ
        g) =
    monoidalComp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (tensorUnit.{v₁, u₁} C (𝒞 := inst)) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (𝟙 (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (monoidalComp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) X (@leftDual.{v₁, u₁} _ inst inst_1 Y _)) Y))
        (tensorUnit.{v₁, u₁} C (𝒞 := inst) ◁ η_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ≫
          η_ (@leftDual.{v₁, u₁} _ inst inst_1 X _) X ▷
            tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y)
        (monoidalComp (Y :=
          tensorObj.{v₁, u₁} (𝒞 := inst) (@leftDual.{v₁, u₁} _ inst inst_1 X _)
            (tensorObj.{v₁, u₁} (𝒞 := inst) (tensorObj.{v₁, u₁} (𝒞 := inst) Y (@leftDual.{v₁, u₁} _ inst inst_1 Y _))
              Y))
          (@leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ f ▷ @leftDual.{v₁, u₁} _ inst inst_1 Y _ ▷ Y)
          (@leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ ε_ (@leftDual.{v₁, u₁} _ inst inst_1 Y _) Y ▷ Y ⊗≫
            @leftDual.{v₁, u₁} _ inst inst_1 X _ ◁ g)))

Tactic:
rw [tensorHom_def]

Premises used: CategoryTheory.MonoidalCategory.tensorHom_def