-- 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 : HasRightDual.{v₁, u₁} X
inst_3 : HasRightDual.{v₁, u₁} Y
inst_4 : HasRightDual.{v₁, u₁} Z
f : X ⟶ Y
g : Y ⟶ Z
simp :
  CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Z _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (Iso.inv (ρ_ (@rightDual.{v₁, u₁} _ inst inst_1 Z _)))
      (@rightDual.{v₁, u₁} _ inst inst_1 Z _ ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
        @rightDual.{v₁, u₁} _ inst inst_1 Z _ ◁
            (f ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _ ≫ g ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
          Iso.inv (α_ (@rightDual.{v₁, u₁} _ inst inst_1 Z _) Z (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ≫
            ε_ Z (@rightDual.{v₁, u₁} _ inst inst_1 Z _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _ ≫
              Iso.hom (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _))) =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Z _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (Iso.inv (ρ_ (@rightDual.{v₁, u₁} _ inst inst_1 Z _)))
      (@rightDual.{v₁, u₁} _ inst inst_1 Z _ ◁ η_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _) ≫
        @rightDual.{v₁, u₁} _ inst inst_1 Z _ ◁
            (g ⊗ₘ
              CategoryStruct.comp (Y :=
                tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Y _)
                  (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
                (Iso.inv (ρ_ (@rightDual.{v₁, u₁} _ inst inst_1 Y _)))
                (@rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ η_ X (@rightDual.{v₁, u₁} _ inst inst_1 X _) ≫
                  @rightDual.{v₁, u₁} _ inst inst_1 Y _ ◁ f ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _ ≫
                    Iso.inv (α_ (@rightDual.{v₁, u₁} _ inst inst_1 Y _) Y (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ≫
                      ε_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _ ≫
                        Iso.hom (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _)))) ≫
          Iso.inv (α_ (@rightDual.{v₁, u₁} _ inst inst_1 Z _) Z (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ≫
            ε_ Z (@rightDual.{v₁, u₁} _ inst inst_1 Z _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _ ≫
              Iso.hom (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _)))
⊢ (f ≫ g)ᘁ =
    CategoryStruct.comp (Y :=
      tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Z _) (tensorUnit.{v₁, u₁} C (𝒞 := inst)))
      (Iso.inv (ρ_ (@rightDual.{v₁, u₁} _ inst inst_1 Z _)))
      (CategoryStruct.comp (Y :=
        tensorObj.{v₁, u₁} (𝒞 := inst) (@rightDual.{v₁, u₁} _ inst inst_1 Z _)
          (tensorObj.{v₁, u₁} (𝒞 := inst) Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _)))
        (@rightDual.{v₁, u₁} _ inst inst_1 Z _ ◁ η_ Y (@rightDual.{v₁, u₁} _ inst inst_1 Y _))
        (@rightDual.{v₁, u₁} _ inst inst_1 Z _ ◁ (g ⊗ₘ fᘁ) ≫
          Iso.inv (α_ (@rightDual.{v₁, u₁} _ inst inst_1 Z _) Z (@rightDual.{v₁, u₁} _ inst inst_1 X _)) ≫
            ε_ Z (@rightDual.{v₁, u₁} _ inst inst_1 Z _) ▷ @rightDual.{v₁, u₁} _ inst inst_1 X _ ≫
              Iso.hom (λ_ (@rightDual.{v₁, u₁} _ inst inst_1 X _))))

Tactic:
simp only [rightAdjointMate, comp_whiskerRight]

Premises used: CategoryTheory.rightAdjointMate, CategoryTheory.MonoidalCategory.comp_whiskerRight