-- Module: Mathlib.Condensed.Discrete.Colimit
Goal:
I : Type u
inst : Category.{u, u} I
inst_1 : IsCofiltered I
F : I ⥤ FintypeCat
c : Cone (F ⋙ toProfinite)
X : Type (u + 1)
hc : IsLimit c
inst_2 : ∀ (i : I), Epi (c.π.app i)
i j : I
fi :
  LocallyConstant
    (↑(((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op i))).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u))) :
      Type u)
    X
fj :
  LocallyConstant
    (↑(((F ⋙ toProfinite).obj (Opposite.unop (Opposite.op j))).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u))) :
      Type u)
    X
h :
  LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi = LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj
k : I
ki : k ⟶ i
kj : k ⟶ j
h_1 : True
x :
  (↑((toProfinite.obj (F.obj k)).toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
    Type u)
refine_2 :
  ∀
    (x_1 :
      (↑((obj (obj (const I) c.pt) k).toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
        Type u))
    (hx :
      (ConcreteCategory.hom (FC :=
              fun (x1 x2 : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
              C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u),
                (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u)))
              (CC := fun (X : CompHausLike fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u)) ↦
              (↑(X.toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) : Type u))
              (c.π.app k) :
            (a :
                (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                  (((const I).obj c.pt).obj k)) →
              (fun (X : CompHausLike fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u)) ↦ (↑X.toTop : Type u))
                ((F ⋙ toProfinite).obj k))
          x_1 =
        x),
    Eq (α := X)
      (((({
                    toFun :=
                      fun
                        (g :
                          LocallyConstant
                            (↑((toProfinite.obj (F.obj i)).toTop (P := fun (X : TopCat.{u}) ↦
                                  TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                              Type u)
                            X) ↦
                      LocallyConstant.comap (TopCat.Hom.hom (toProfinite.map (F.map ki)).hom) g } :
                  (a : LocallyConstant (↑(toProfinite.obj (F.obj i)).toTop : Type u) X) →
                    LocallyConstant (↑(toProfinite.obj (F.obj k)).toTop : Type u) X)
                fi :
              LocallyConstant
                (↑((toProfinite.obj (F.obj k)).toTop (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                  Type u)
                X) :
            (a : (↑(toProfinite.obj (F.obj k)).toTop : Type u)) → X)
          x :
        X)
      (((({
                    toFun :=
                      fun
                        (g :
                          LocallyConstant
                            (↑((toProfinite.obj (F.obj j)).toTop (P := fun (X : TopCat.{u}) ↦
                                  TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                              Type u)
                            X) ↦
                      LocallyConstant.comap (TopCat.Hom.hom (toProfinite.map (F.map kj)).hom) g } :
                  (a : LocallyConstant (↑(toProfinite.obj (F.obj j)).toTop : Type u) X) →
                    LocallyConstant (↑(toProfinite.obj (F.obj k)).toTop : Type u) X)
                fj :
              LocallyConstant
                (↑((toProfinite.obj (F.obj k)).toTop (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                  Type u)
                X) :
            (a : (↑(toProfinite.obj (F.obj k)).toTop : Type u)) → X)
          x :
        X)
⊢ Eq (α := X)
    (((({
                  toFun :=
                    fun
                      (g :
                        LocallyConstant
                          (↑((toProfinite.obj (F.obj i)).toTop (P := fun (X : TopCat.{u}) ↦
                                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u)
                          X) ↦
                    LocallyConstant.comap (TopCat.Hom.hom (toProfinite.map (F.map ki)).hom) g } :
                (a : LocallyConstant (↑(toProfinite.obj (F.obj i)).toTop : Type u) X) →
                  LocallyConstant (↑(toProfinite.obj (F.obj k)).toTop : Type u) X)
              fi :
            LocallyConstant
              (↑((toProfinite.obj (F.obj k)).toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                Type u)
              X) :
          (a : (↑(toProfinite.obj (F.obj k)).toTop : Type u)) → X)
        x :
      X)
    (((({
                  toFun :=
                    fun
                      (g :
                        LocallyConstant
                          (↑((toProfinite.obj (F.obj j)).toTop (P := fun (X : TopCat.{u}) ↦
                                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                            Type u)
                          X) ↦
                    LocallyConstant.comap (TopCat.Hom.hom (toProfinite.map (F.map kj)).hom) g } :
                (a : LocallyConstant (↑(toProfinite.obj (F.obj j)).toTop : Type u) X) →
                  LocallyConstant (↑(toProfinite.obj (F.obj k)).toTop : Type u) X)
              fj :
            LocallyConstant
              (↑((toProfinite.obj (F.obj k)).toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u))) :
                Type u)
              X) :
          (a : (↑(toProfinite.obj (F.obj k)).toTop : Type u)) → X)
        x :
      X)

Tactic:
obtain ⟨x, hx⟩ := ((Profinite.epi_iff_surjective (c.π.app k)).mp inferInstance) x

Premises used: CategoryTheory.Limits.Cone.π, Profinite.epi_iff_surjective, inferInstance, Iff.mp, CategoryTheory.NatTrans.app