-- Lean 4 proof step from Mathlib.Condensed.Discrete.Colimit (hypothesis)
Goal:
F : ℕᵒᵖ ⥤ FintypeCat
c : Cone (F ⋙ toLightProfinite)
X : Type u
hc : IsLimit c
inst : ∀ (i : ℕᵒᵖ), Epi (c.π.app i)
i j : ℕᵒᵖ
fi :
  LocallyConstant
    (↑(((F ⋙ toLightProfinite).obj (Opposite.unop (Opposite.op i))).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u)
    X
fj :
  LocallyConstant
    (↑(((F ⋙ toLightProfinite).obj (Opposite.unop (Opposite.op j))).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{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 : ℕᵒᵖ
ki : k ⟶ i
kj : k ⟶ j
h_1 : True
x_1 :
  (↑((toLightProfinite.obj (F.obj k)).toTop (P := fun (X : TopCat.{u}) ↦
        TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
    Type u)
x :
  (↑((obj (obj (const ℕᵒᵖ) c.pt) k).toTop (P := fun (X : TopCat.{u}) ↦
        TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
    Type u)
hx :
  (ConcreteCategory.hom (FC :=
          fun
            (x1 x2 :
              CompHausLike fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
          C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
              Type u),
            (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
              Type u)))
          (CC :=
          fun
            (X :
              CompHausLike fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
          (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
            Type u))
          (c.π.app k) :
        (a :
            (fun
                  (X :
                    CompHausLike fun (X : TopCat) ↦
                      TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                (↑X.toTop : Type u))
              (((const ℕᵒᵖ).obj c.pt).obj k)) →
          (fun
                (X :
                  CompHausLike fun (X : TopCat) ↦
                    TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
              (↑X.toTop : Type u))
            ((F ⋙ toLightProfinite).obj k))
      x =
    x_1
⊢ Eq (α := X)
    ((fi : (a : (↑((F ⋙ toLightProfinite).obj (Opposite.unop (Opposite.op i))).toTop : Type u)) → X)
        ((ConcreteCategory.hom (FC :=
                fun
                  (x1 x2 :
                    CompHausLike fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                        TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                    Type u),
                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                        TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                    Type u)))
                (CC :=
                fun
                  (X :
                    CompHausLike fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                  Type u))
                (c.π.app k ≫ (F ⋙ toLightProfinite).map ki) :
              (a :
                  (fun
                        (X :
                          CompHausLike fun (X : TopCat) ↦
                            TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                      (↑X.toTop : Type u))
                    (((const ℕᵒᵖ).obj c.pt).obj k)) →
                (fun
                      (X :
                        CompHausLike fun (X : TopCat) ↦
                          TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                    (↑X.toTop : Type u))
                  ((F ⋙ toLightProfinite).obj (Opposite.unop (Opposite.op i))))
            x :
          (fun
                (X :
                  CompHausLike fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
              (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                Type u))
            ((F ⋙ toLightProfinite).obj (Opposite.unop (Opposite.op i)))) :
      X)
    ((fj : (a : (↑((F ⋙ toLightProfinite).obj (Opposite.unop (Opposite.op j))).toTop : Type u)) → X)
        ((ConcreteCategory.hom (FC :=
                fun
                  (x1 x2 :
                    CompHausLike fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
                C((↑(x1.toTop (P := fun (X : TopCat.{u}) ↦
                        TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                    Type u),
                  (↑(x2.toTop (P := fun (X : TopCat.{u}) ↦
                        TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                    Type u)))
                (CC :=
                fun
                  (X :
                    CompHausLike fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
                (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                      TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                  Type u))
                (c.π.app k ≫ (F ⋙ toLightProfinite).map kj) :
              (a :
                  (fun
                        (X :
                          CompHausLike fun (X : TopCat) ↦
                            TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                      (↑X.toTop : Type u))
                    (((const ℕᵒᵖ).obj c.pt).obj k)) →
                (fun
                      (X :
                        CompHausLike fun (X : TopCat) ↦
                          TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u)) ↦
                    (↑X.toTop : Type u))
                  ((F ⋙ toLightProfinite).obj (Opposite.unop (Opposite.op j))))
            x :
          (fun
                (X :
                  CompHausLike fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) ↦
              (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
                Type u))
            ((F ⋙ toLightProfinite).obj (Opposite.unop (Opposite.op j)))) :
      X)
have h₁ : Eq (α := X)
  ((LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi :
        (a : (↑(((const ℕᵒᵖ).obj c.pt).obj i).toTop : Type u)) → X)
      x :
    X)
  ((LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj :
        (a : (↑(((const ℕᵒᵖ).obj c.pt).obj i).toTop : Type u)) → X)
      x :
    X)
Tactic: have h := LocallyConstant.congr_fun h x