-- Mathlib/CategoryTheory/Closed/Functor.lean
theorem frobeniusMorphism_mate (h : L ⊣ F) (A : C) :
    conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h)
        (frobeniusMorphism F h A).natTrans = (expComparison F A).natTrans

Proof state:
case e_w.h.h.inv_hom_id
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
  ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
        ((mateEquiv h h)
          (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
    (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
      (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
B : C
ηlemma :
  h.unit.app (F.obj A ⊗ F.obj B) ≫
      lift ((L ⋙ F).map (fst (F.obj A) (F.obj B))) ((L ⋙ F).map (snd (F.obj A) (F.obj B))) =
    h.unit.app (F.obj A) ⊗ h.unit.app (F.obj B)
⊢ (h.unit.app (F.obj A) ⊗ h.unit.app (F.obj B)) ≫ (F.map (h.counit.app A) ⊗ F.map (h.counit.app B)) =
    𝟙 (F.obj A ⊗ F.obj B)

Tactic:
ext <;> simp

Result:
no goals

Proof state:
case e_w.h
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
  ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
        ((mateEquiv h h)
          (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
    (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
      (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
⊢ (mateEquiv h h) (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))) =
    (prodComparisonNatIso F A).inv

Tactic:
ext B

Result:
case e_w.h.h
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
  ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
        ((mateEquiv h h)
          (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
    (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
      (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
B : C
⊢ ((mateEquiv h h)
            (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))).natTrans.app
      B =
    (natTrans (prodComparisonNatIso F A).inv).app B

Proof state:
case e_w.h.h.inv_hom_id
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
  ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
        ((mateEquiv h h)
          (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
    (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
      (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
B : C
⊢ (h.unit.app (F.obj A ⊗ F.obj B) ≫
        F.map (prodComparison L (F.obj A) (F.obj B)) ≫
          F.map (h.counit.app A ▷ L.obj (F.obj B)) ≫ F.map (A ◁ h.counit.app B)) ≫
      prodComparison F A B =
    𝟙 (F.obj A ⊗ F.obj B)

Tactic:
simp only [assoc]

Result:
case e_w.h.h.inv_hom_id
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
  ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
        ((mateEquiv h h)
          (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
    (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
      (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
B : C
⊢ h.unit.app (F.obj A ⊗ F.obj B) ≫
      F.map (prodComparison L (F.obj A) (F.obj B)) ≫
        F.map (h.counit.app A ▷ L.obj (F.obj B)) ≫ F.map (A ◁ h.counit.app B) ≫ prodComparison F A B =
    𝟙 (F.obj A ⊗ F.obj B)

Proof state:
case e_w.h.h
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
  ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
        ((mateEquiv h h)
          (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
    (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
      (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
B : C
⊢ h.unit.app (F.obj A ⊗ F.obj B) ≫
      F.map (prodComparison L (F.obj A) (F.obj B) ≫ h.counit.app A ▷ L.obj (F.obj B) ≫ A ◁ h.counit.app B) =
    inv (prodComparison F A B)

Tactic:
simp only [Functor.map_comp]

Result:
case e_w.h.h
C : Type u
inst✝⁶ : Category.{v, u} C
D : Type u'
inst✝⁵ : Category.{v, u'} D
inst✝⁴ : ChosenFiniteProducts C
inst✝³ : ChosenFiniteProducts D
F : C ⥤ D
L : D ⥤ C
inst✝² : CartesianClosed C
inst✝¹ : CartesianClosed D
inst✝ : Limits.PreservesLimitsOfShape (Discrete Limits.WalkingPair) F
h : L ⊣ F
A : C
conjeq :
  ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A)))
        ((mateEquiv h h)
          (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A))))).natTrans =
    (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
      (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))
B : C
⊢ h.unit.app (F.obj A ⊗ F.obj B) ≫
      F.map (prodComparison L (F.obj A) (F.obj B)) ≫
        F.map (h.counit.app A ▷ L.obj (F.obj B)) ≫ F.map (A ◁ h.counit.app B) =
    inv (prodComparison F A B)