-- 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
⊢ h.unit.app (F.obj A ⊗ F.obj B) ≫
      (prodComparison (L ⋙ F) (F.obj A) (F.obj B) ≫ F.map (h.counit.app A) ▷ F.obj (L.obj (F.obj B))) ≫
        F.obj A ◁ F.map (h.counit.app 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) ≫
      prodComparison (L ⋙ F) (F.obj A) (F.obj B) ≫
        F.map (h.counit.app A) ▷ F.obj (L.obj (F.obj B)) ≫ F.obj A ◁ F.map (h.counit.app B) =
    𝟙 (F.obj A ⊗ F.obj B)

Proof state:
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)))
⊢ (conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h))
      (natTrans (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))) =
    ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv).natTrans

Tactic:
rw [← conjeq]

Result:
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 (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 =
    ((mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv).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.obj (L.obj (F.obj B))) ≫
      F.obj A ◁ F.map (h.counit.app B) =
    𝟙 (F.obj A ⊗ F.obj B)

Tactic:
simp only [Functor.id_obj, Functor.comp_obj, assoc, ← whisker_exchange, ← tensorHom_def']

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
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)