-- 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
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
⊢ 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))
have 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))
Tactic: refine Types.FilteredColimit.isColimitOf _ _ ?_ ?_