-- Module: Mathlib.CategoryTheory.Galois.Prorepresentability
Goal:
C : Type u₁
inst : Category.{u₂, u₁} C
inst_1 : GaloisCategory.{u₁, u₂} C
F : C ⥤ FintypeCat
inst_2 : FiberFunctor F
X A : C
a : (F.obj A).obj
isGalois : IsGalois.{u₁, u₂} A
B : C
b : (F.obj B).obj
isGalois_1 : IsGalois.{u₁, u₂} B
u : A ⟶ X
v : B ⟶ X
refine_2 :
  ∀
    (h :
      Eq (α := (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        ((@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.map u) :
              (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj A)) →
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
            a :
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
        ((@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.map v) :
              (a : (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj B)) →
                (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))
            b :
          (fun (X : ObjectProperty.FullSubcategory Finite) ↦ X.obj) (F.obj X))),
    ∃ (k : (PointedGaloisObject F)ᵒᵖ) (f :
      Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F } ⟶ k) (g :
      Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F } ⟶ k),
      Eq (α :=
        (fun (X : Type u₂) ↦ X)
          (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (map ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) f) :
              (a :
                  (fun (X : Type u₂) ↦ X)
                    ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                      (Opposite.op { obj := A, pt := a, isGalois := isGalois }))) →
                (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj k))
            u :
          (fun (X : Type u₂) ↦ X)
            (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (map ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) g) :
              (a :
                  (fun (X : Type u₂) ↦ X)
                    ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                      (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 }))) →
                (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj k))
            v :
          (fun (X : Type u₂) ↦ X)
            (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) k))
⊢ Eq (α :=
      (fun (X : Type u₂) ↦ X)
        (Functor.obj
          (Functor.obj (const (PointedGaloisObject F)ᵒᵖ)
            (Cocone.pt (J := (PointedGaloisObject F)ᵒᵖ) (F :=
              (Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
              (mapCocone (Functor.obj (evaluation C (Type u₂)) X) (cocone F))))
          (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app (Cocone.ι (mapCocone (Functor.obj (evaluation C (Type u₂)) X) (cocone F)))
                (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })) :
            (a_1 :
                (fun (X : Type u₂) ↦ X)
                  ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                    (Opposite.op { obj := A, pt := a, isGalois := isGalois }))) →
              (fun (X : Type u₂) ↦ X)
                (((const (PointedGaloisObject F)ᵒᵖ).obj (((evaluation C (Type u₂)).obj X).mapCocone (cocone F)).pt).obj
                  (Opposite.op { obj := A, pt := a, isGalois := isGalois })))
          u :
        (fun (X : Type u₂) ↦ X)
          (Functor.obj
            (Functor.obj (const (PointedGaloisObject F)ᵒᵖ)
              (Cocone.pt (J := (PointedGaloisObject F)ᵒᵖ) (F :=
                (Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
                (mapCocone (Functor.obj (evaluation C (Type u₂)) X) (cocone F))))
            (Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F })))
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app (Cocone.ι (mapCocone (Functor.obj (evaluation C (Type u₂)) X) (cocone F)))
                (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F })) :
            (a :
                (fun (X : Type u₂) ↦ X)
                  ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                    (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 }))) →
              (fun (X : Type u₂) ↦ X)
                (((const (PointedGaloisObject F)ᵒᵖ).obj (((evaluation C (Type u₂)).obj X).mapCocone (cocone F)).pt).obj
                  (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 })))
          v :
        (fun (X : Type u₂) ↦ X)
          (Functor.obj
            (Functor.obj (const (PointedGaloisObject F)ᵒᵖ)
              (Cocone.pt (J := (PointedGaloisObject F)ᵒᵖ) (F :=
                (Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X)
                (mapCocone (Functor.obj (evaluation C (Type u₂)) X) (cocone F))))
            (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F }))) →
    ∃ (k : (PointedGaloisObject F)ᵒᵖ) (f :
      Opposite.op { obj := A, pt := a, isGalois := isGalois : PointedGaloisObject F } ⟶ k) (g :
      Opposite.op { obj := B, pt := b, isGalois := isGalois_1 : PointedGaloisObject F } ⟶ k),
      Eq (α :=
        (fun (X : Type u₂) ↦ X)
          (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (map ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) f) :
              (a :
                  (fun (X : Type u₂) ↦ X)
                    ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                      (Opposite.op { obj := A, pt := a, isGalois := isGalois }))) →
                (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj k))
            u :
          (fun (X : Type u₂) ↦ X)
            (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) k))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (map ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) g) :
              (a :
                  (fun (X : Type u₂) ↦ X)
                    ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj
                      (Opposite.op { obj := B, pt := b, isGalois := isGalois_1 }))) →
                (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj k))
            v :
          (fun (X : Type u₂) ↦ X)
            (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) k))

Tactic:
(h : F.map u a = F.map v b)