-- Lean 4 proof step from Mathlib.CategoryTheory.Galois.Prorepresentability (goal)
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : C ⥤ FintypeCat
inst_2 : FiberFunctor F
f : End F
A : PointedGaloisObject F
⊢ Eq.{u₂ + 1} (α := Functor.obj (incl F ⋙ F ⋙ FintypeCat.incl) A)
    ((↑((endEquivSectionsFibers F : (a : End F) → (↑(incl F ⋙ F ⋙ FintypeCat.incl).sections : Type (max u₁ u₂))) f) :
          (j : PointedGaloisObject F) → (incl F ⋙ F ⋙ FintypeCat.incl).obj j)
        A :
      Functor.obj (incl F ⋙ F ⋙ FintypeCat.incl) A)
    ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
            (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
              (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
            _ _ _ (f.app A.obj : F.obj A.obj ⟶ F.obj A.obj) :
          (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A.obj)) →
            (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A.obj))
        A.pt :
      (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A.obj))
have dsimp : Eq (α := (F.obj A.obj).obj)
  (ULift.down.{u₁, u₂}
      ((↑((Types.limitEquivSections (incl F ⋙ (F ⋙ FintypeCat.incl) ⋙ uliftFunctor.{u₁, u₂}) :
                (a : limit ((incl F ⋙ F ⋙ FintypeCat.incl) ⋙ uliftFunctor.{u₁, u₂})) →
                  (↑((incl F ⋙ F ⋙ FintypeCat.incl) ⋙ uliftFunctor.{u₁, u₂}).sections : Type (max u₁ u₂)))
              (((ConcreteCategory.hom (FC := TypeCat.Fun)
                      (Iso.hom (colimitCoyonedaHomIsoLimit' (incl F) (F ⋙ FintypeCat.incl))) :
                    TypeCat.Fun
                      (colimit (J := (PointedGaloisObject F)ᵒᵖ) (Functor.op (incl F) ⋙ coyoneda) ⟶ F ⋙ FintypeCat.incl)
                      (limit (J := PointedGaloisObject F) ((incl F ⋙ F ⋙ FintypeCat.incl) ⋙ uliftFunctor.{u₁, u₂}))) :
                  (a : colimit ((incl F).op ⋙ coyoneda) ⟶ F ⋙ FintypeCat.incl) →
                    limit ((incl F ⋙ F ⋙ FintypeCat.incl) ⋙ uliftFunctor.{u₁, u₂}))
                (CategoryStruct.comp (Y := Cocone.pt (J := (PointedGaloisObject F)ᵒᵖ) (cocone F))
                  (Iso.hom (colimit.isoColimitCocone { cocone := cocone F, isColimit := isColimit F }) :
                    colimit (J := (PointedGaloisObject F)ᵒᵖ) (Functor.op (incl F) ⋙ coyoneda) ⟶
                      Cocone.pt (J := (PointedGaloisObject F)ᵒᵖ) (cocone F))
                  ((FullyFaithful.homEquiv ((FullyFaithful.ofFullyFaithful FintypeCat.incl).whiskeringRight C) :
                      (a : End F) → End (F ⋙ FintypeCat.incl))
                    f)))) :
          (j : PointedGaloisObject F) → ULift.{u₁, u₂} (F.obj j.obj).obj)
        A) :
    (F.obj A.obj).obj)
  ((@ConcreteCategory.hom _ (ObjectProperty.FullSubcategory.category Finite)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦ TypeCat.Fun X.obj Y.obj)
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj)
          (fun (X Y : ObjectProperty.FullSubcategory Finite) ↦
            (TypeCat.instFunLikeFun : FunLike (TypeCat.Fun X.obj Y.obj) X.obj Y.obj))
          _ _ _ (f.app A.obj : F.obj A.obj ⟶ F.obj A.obj) :
        (a : (F.obj A.obj).obj) → (F.obj A.obj).obj)
      A.pt :
    (F.obj A.obj).obj)
Tactic: dsimp [endEquivSectionsFibers, Types.sectionsEquiv]