-- 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) ≫
      lift ((L ⋙ F).map (fst (F.obj A) (F.obj B))) ((L ⋙ F).map (snd (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:
have ηlemma : (h.unit.app (F.obj A ⊗ F.obj B) ≫
  lift ((L ⋙ F).map (fst _ _)) ((L ⋙ F).map (snd _ _))) =
    (h.unit.app (F.obj A)) ⊗ (h.unit.app (F.obj B)) := by
  ext <;> simp

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 ⊗ F.obj B) ≫
      lift ((L ⋙ F).map (fst (F.obj A) (F.obj B))) ((L ⋙ F).map (snd (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:
case e_w
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)))) =
    (mateEquiv (exp.adjunction A) (exp.adjunction (F.obj A))) (prodComparisonNatIso F A).inv

Tactic:
apply congr_arg

Result:
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

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
⊢ (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:
have conjeq := iterated_mateEquiv_conjugateEquiv h h
  (exp.adjunction (F.obj A)) (exp.adjunction A)
  (prodComparisonNatTrans L (F.obj A) ≫ whiskerLeft L ((curriedTensor C).map (h.counit.app A)))

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)))
⊢ (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