-- 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 : C
refine_1 :
  ∀
    (x :
      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))),
    ∃ (i : (PointedGaloisObject F)ᵒᵖ) (xi :
      (fun (X : Type u₂) ↦ X)
        (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) i)),
      x =
        (ConcreteCategory.hom (FC := TypeCat.Fun)
              (NatTrans.app (Cocone.ι (mapCocone (Functor.obj (evaluation C (Type u₂)) X) (cocone F))) i) :
            (a : (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj i)) →
              (fun (X : Type u₂) ↦ X)
                (((const (PointedGaloisObject F)ᵒᵖ).obj (((evaluation C (Type u₂)).obj X).mapCocone (cocone F)).pt).obj
                  i))
          xi
refine_2 :
  ∀ (i j : (PointedGaloisObject F)ᵒᵖ)
    (xi :
      (fun (X : Type u₂) ↦ X)
        (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) i))
    (xj :
      (fun (X : Type u₂) ↦ X)
        (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) j)),
    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))))
            i))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app (Cocone.ι (mapCocone (Functor.obj (evaluation C (Type u₂)) X) (cocone F))) i) :
              (a : (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj i)) →
                (fun (X : Type u₂) ↦ X)
                  (((const (PointedGaloisObject F)ᵒᵖ).obj
                        (((evaluation C (Type u₂)).obj X).mapCocone (cocone F)).pt).obj
                    i))
            xi :
          (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))))
              i))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app (Cocone.ι (mapCocone (Functor.obj (evaluation C (Type u₂)) X) (cocone F))) j) :
              (a : (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj j)) →
                (fun (X : Type u₂) ↦ X)
                  (((const (PointedGaloisObject F)ᵒᵖ).obj
                        (((evaluation C (Type u₂)).obj X).mapCocone (cocone F)).pt).obj
                    j))
            xj :
          (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))))
              j)) →
      ∃ (k : (PointedGaloisObject F)ᵒᵖ) (f : i ⟶ k) (g : j ⟶ 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 i)) →
                  (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj k))
              xi :
            (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 j)) →
                  (fun (X : Type u₂) ↦ X) ((((incl F).op ⋙ coyoneda) ⋙ (evaluation C (Type u₂)).obj X).obj k))
              xj :
            (fun (X : Type u₂) ↦ X)
              (Functor.obj ((Functor.op (incl F) ⋙ coyoneda) ⋙ Functor.obj (evaluation C (Type u₂)) X) k))
⊢ IsColimit (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))

Tactic:
refine Types.FilteredColimit.isColimitOf _ _ ?_ ?_

Premises used: CategoryTheory.Limits.Types.FilteredColimit.isColimitOf