-- Module: Mathlib.Condensed.Discrete.Colimit
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
refine_2 :
  ∀ (k : ℕᵒᵖ) (ki : k ⟶ i) (kj : k ⟶ j),
    True →
      ∃ (k : ℕᵒᵖᵒᵖ) (f : Opposite.op i ⟶ k) (g : Opposite.op j ⟶ k),
        Eq (α := (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).map f) :
                (a :
                    (fun (X : Type u) ↦ X)
                      (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op i))) →
                  (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
              fi :
            (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).map g) :
                (a :
                    (fun (X : Type u) ↦ X)
                      (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op j))) →
                  (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
              fj :
            (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
⊢ ∃ (k : ℕᵒᵖᵒᵖ) (f : Opposite.op i ⟶ k) (g : Opposite.op j ⟶ k),
    Eq (α := (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).map f) :
            (a : (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op i))) →
              (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
          fi :
        (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).map g) :
            (a : (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op j))) →
              (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))
          fj :
        (fun (X : Type u) ↦ X) (((F ⋙ toLightProfinite).op ⋙ locallyConstantPresheaf X).obj k))

Tactic:
obtain ⟨k, ki, kj, _⟩ := IsCofilteredOrEmpty.cone_objs i j

Premises used: CategoryTheory.IsCofilteredOrEmpty.cone_objs