-- Mathlib/CategoryTheory/Adjunction/Mates.lean
theorem conjugateEquiv_comp (α : L₂ ⟶ L₁) (β : L₃ ⟶ L₂) :
    conjugateEquiv adj₁ adj₂ α ≫ conjugateEquiv adj₂ adj₃ β =
      conjugateEquiv adj₁ adj₃ (β ≫ α)

Proof state:
case w.h
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
L₁ L₂ L₃ : C ⥤ D
R₁ R₂ R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : L₂ ⟶ L₁
β : L₃ ⟶ L₂
d : D
⊢ (𝟙 (R₁.obj d) ≫
        (adj₂.unit.app (R₁.obj d) ≫
            R₂.map (𝟙 (L₂.obj (R₁.obj d)) ≫ α.app (R₁.obj d) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫ R₂.map (adj₁.counit.app d)) ≫
          𝟙 (R₂.obj d)) ≫
      𝟙 (R₂.obj d) ≫
        (adj₃.unit.app (R₂.obj d) ≫
            R₃.map (𝟙 (L₃.obj (R₂.obj d)) ≫ β.app (R₂.obj d) ≫ 𝟙 (L₂.obj (R₂.obj d))) ≫ R₃.map (adj₂.counit.app d)) ≫
          𝟙 (R₃.obj d) =
    𝟙 (R₁.obj d) ≫
      (adj₃.unit.app (R₁.obj d) ≫
          R₃.map (𝟙 (L₃.obj (R₁.obj d)) ≫ (β.app (R₁.obj d) ≫ α.app (R₁.obj d)) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫
            R₃.map (adj₁.counit.app d)) ≫
        𝟙 (R₃.obj d)

Tactic:
have vcomp := mateEquiv_vcomp adj₁ adj₂ adj₃
  (L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv)
  (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)

Result:
case w.h
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
L₁ L₂ L₃ : C ⥤ D
R₁ R₂ R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : L₂ ⟶ L₁
β : L₃ ⟶ L₂
d : D
vcomp :
  (mateEquiv adj₁ adj₃) ((L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ₕ L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv) =
    (mateEquiv adj₁ adj₂) (L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ᵥ
      (mateEquiv adj₂ adj₃) (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)
⊢ (𝟙 (R₁.obj d) ≫
        (adj₂.unit.app (R₁.obj d) ≫
            R₂.map (𝟙 (L₂.obj (R₁.obj d)) ≫ α.app (R₁.obj d) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫ R₂.map (adj₁.counit.app d)) ≫
          𝟙 (R₂.obj d)) ≫
      𝟙 (R₂.obj d) ≫
        (adj₃.unit.app (R₂.obj d) ≫
            R₃.map (𝟙 (L₃.obj (R₂.obj d)) ≫ β.app (R₂.obj d) ≫ 𝟙 (L₂.obj (R₂.obj d))) ≫ R₃.map (adj₂.counit.app d)) ≫
          𝟙 (R₃.obj d) =
    𝟙 (R₁.obj d) ≫
      (adj₃.unit.app (R₁.obj d) ≫
          R₃.map (𝟙 (L₃.obj (R₁.obj d)) ≫ (β.app (R₁.obj d) ≫ α.app (R₁.obj d)) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫
            R₃.map (adj₁.counit.app d)) ≫
        𝟙 (R₃.obj d)

Proof state:
case w.h
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
L₁ L₂ L₃ : C ⥤ D
R₁ R₂ R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : L₂ ⟶ L₁
β : L₃ ⟶ L₂
d : D
⊢ ((conjugateEquiv adj₁ adj₂) α ≫ (conjugateEquiv adj₂ adj₃) β).app d = ((conjugateEquiv adj₁ adj₃) (β ≫ α)).app d

Tactic:
dsimp [conjugateEquiv, mateEquiv]

Result:
case w.h
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
L₁ L₂ L₃ : C ⥤ D
R₁ R₂ R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : L₂ ⟶ L₁
β : L₃ ⟶ L₂
d : D
⊢ (𝟙 (R₁.obj d) ≫
        (adj₂.unit.app (R₁.obj d) ≫
            R₂.map (𝟙 (L₂.obj (R₁.obj d)) ≫ α.app (R₁.obj d) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫ R₂.map (adj₁.counit.app d)) ≫
          𝟙 (R₂.obj d)) ≫
      𝟙 (R₂.obj d) ≫
        (adj₃.unit.app (R₂.obj d) ≫
            R₃.map (𝟙 (L₃.obj (R₂.obj d)) ≫ β.app (R₂.obj d) ≫ 𝟙 (L₂.obj (R₂.obj d))) ≫ R₃.map (adj₂.counit.app d)) ≫
          𝟙 (R₃.obj d) =
    𝟙 (R₁.obj d) ≫
      (adj₃.unit.app (R₁.obj d) ≫
          R₃.map (𝟙 (L₃.obj (R₁.obj d)) ≫ (β.app (R₁.obj d) ≫ α.app (R₁.obj d)) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫
            R₃.map (adj₁.counit.app d)) ≫
        𝟙 (R₃.obj d)

Proof state:
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
L₁ L₂ L₃ : C ⥤ D
R₁ R₂ R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : L₂ ⟶ L₁
β : L₃ ⟶ L₂
⊢ (conjugateEquiv adj₁ adj₂) α ≫ (conjugateEquiv adj₂ adj₃) β = (conjugateEquiv adj₁ adj₃) (β ≫ α)

Tactic:
ext d

Result:
case w.h
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
L₁ L₂ L₃ : C ⥤ D
R₁ R₂ R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : L₂ ⟶ L₁
β : L₃ ⟶ L₂
d : D
⊢ ((conjugateEquiv adj₁ adj₂) α ≫ (conjugateEquiv adj₂ adj₃) β).app d = ((conjugateEquiv adj₁ adj₃) (β ≫ α)).app d

Proof state:
case w.h
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
L₁ L₂ L₃ : C ⥤ D
R₁ R₂ R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : L₂ ⟶ L₁
β : L₃ ⟶ L₂
d : D
vcomp :
  (mateEquiv adj₁ adj₃) ((L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ₕ L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv) =
    (mateEquiv adj₁ adj₂) (L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ᵥ
      (mateEquiv adj₂ adj₃) (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)
vcompd :
  ((mateEquiv adj₁ adj₃)
          ((L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ₕ L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)).app
      d =
    ((mateEquiv adj₁ adj₂) (L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ᵥ
          (mateEquiv adj₂ adj₃) (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)).app
      d
⊢ (𝟙 (R₁.obj d) ≫
        (adj₂.unit.app (R₁.obj d) ≫
            R₂.map (𝟙 (L₂.obj (R₁.obj d)) ≫ α.app (R₁.obj d) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫ R₂.map (adj₁.counit.app d)) ≫
          𝟙 (R₂.obj d)) ≫
      𝟙 (R₂.obj d) ≫
        (adj₃.unit.app (R₂.obj d) ≫
            R₃.map (𝟙 (L₃.obj (R₂.obj d)) ≫ β.app (R₂.obj d) ≫ 𝟙 (L₂.obj (R₂.obj d))) ≫ R₃.map (adj₂.counit.app d)) ≫
          𝟙 (R₃.obj d) =
    𝟙 (R₁.obj d) ≫
      (adj₃.unit.app (R₁.obj d) ≫
          R₃.map (𝟙 (L₃.obj (R₁.obj d)) ≫ (β.app (R₁.obj d) ≫ α.app (R₁.obj d)) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫
            R₃.map (adj₁.counit.app d)) ≫
        𝟙 (R₃.obj d)

Tactic:
dsimp [mateEquiv, vComp, vComp] at vcompd

Result:
case w.h
C : Type u₁
D : Type u₂
inst✝¹ : Category.{v₁, u₁} C
inst✝ : Category.{v₂, u₂} D
L₁ L₂ L₃ : C ⥤ D
R₁ R₂ R₃ : D ⥤ C
adj₁ : L₁ ⊣ R₁
adj₂ : L₂ ⊣ R₂
adj₃ : L₃ ⊣ R₃
α : L₂ ⟶ L₁
β : L₃ ⟶ L₂
d : D
vcomp :
  (mateEquiv adj₁ adj₃) ((L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ₕ L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv) =
    (mateEquiv adj₁ adj₂) (L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ᵥ
      (mateEquiv adj₂ adj₃) (L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv)
vcompd :
  adj₃.unit.app (R₁.obj d) ≫
      R₃.map
          (((L₂.leftUnitor.hom ≫ α ≫ L₁.rightUnitor.inv) ≫ₕ L₃.leftUnitor.hom ≫ β ≫ L₂.rightUnitor.inv).natTrans.app
            (R₁.obj d)) ≫
        R₃.map (adj₁.counit.app d) =
    𝟙 (R₁.obj d) ≫
      (adj₂.unit.app (R₁.obj d) ≫
          R₂.map (𝟙 (L₂.obj (R₁.obj d)) ≫ α.app (R₁.obj d) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫ R₂.map (adj₁.counit.app d)) ≫
        𝟙 (R₂.obj d) ≫
          (adj₃.unit.app (R₂.obj d) ≫
              R₃.map (𝟙 (L₃.obj (R₂.obj d)) ≫ β.app (R₂.obj d) ≫ 𝟙 (L₂.obj (R₂.obj d))) ≫ R₃.map (adj₂.counit.app d)) ≫
            𝟙 (R₃.obj d)
⊢ (𝟙 (R₁.obj d) ≫
        (adj₂.unit.app (R₁.obj d) ≫
            R₂.map (𝟙 (L₂.obj (R₁.obj d)) ≫ α.app (R₁.obj d) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫ R₂.map (adj₁.counit.app d)) ≫
          𝟙 (R₂.obj d)) ≫
      𝟙 (R₂.obj d) ≫
        (adj₃.unit.app (R₂.obj d) ≫
            R₃.map (𝟙 (L₃.obj (R₂.obj d)) ≫ β.app (R₂.obj d) ≫ 𝟙 (L₂.obj (R₂.obj d))) ≫ R₃.map (adj₂.counit.app d)) ≫
          𝟙 (R₃.obj d) =
    𝟙 (R₁.obj d) ≫
      (adj₃.unit.app (R₁.obj d) ≫
          R₃.map (𝟙 (L₃.obj (R₁.obj d)) ≫ (β.app (R₁.obj d) ≫ α.app (R₁.obj d)) ≫ 𝟙 (L₁.obj (R₁.obj d))) ≫
            R₃.map (adj₁.counit.app d)) ≫
        𝟙 (R₃.obj d)