-- 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 : I
refine_2 :
  ∀ (j : I) (xi : (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op i)))
    (xj : (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op j))),
    Eq (α :=
        (fun (X : Type (u + 1)) ↦ X)
          (obj (obj (const Iᵒᵖ) ((locallyConstantPresheaf X).mapCocone c.op).pt) (Opposite.op i)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (((locallyConstantPresheaf X).mapCocone c.op).ι.app (Opposite.op i)) :
              (a :
                  (fun (X : Type (u + 1)) ↦ X)
                    (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op i))) →
                (fun (X : Type (u + 1)) ↦ X)
                  (((const Iᵒᵖ).obj ((locallyConstantPresheaf X).mapCocone c.op).pt).obj (Opposite.op i)))
            xi :
          (fun (X : Type (u + 1)) ↦ X)
            (obj (obj (const Iᵒᵖ) ((locallyConstantPresheaf X).mapCocone c.op).pt) (Opposite.op i)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (((locallyConstantPresheaf X).mapCocone c.op).ι.app (Opposite.op j)) :
              (a :
                  (fun (X : Type (u + 1)) ↦ X)
                    (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op j))) →
                (fun (X : Type (u + 1)) ↦ X)
                  (((const Iᵒᵖ).obj ((locallyConstantPresheaf X).mapCocone c.op).pt).obj (Opposite.op j)))
            xj :
          (fun (X : Type (u + 1)) ↦ X)
            (obj (obj (const Iᵒᵖ) ((locallyConstantPresheaf X).mapCocone c.op).pt) (Opposite.op j))) →
      ∃ (k : Iᵒᵖ) (f : Opposite.op i ⟶ k) (g : Opposite.op j ⟶ k),
        Eq (α := (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).map f) :
                (a :
                    (fun (X : Type (u + 1)) ↦ X)
                      (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op i))) →
                  (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
              xi :
            (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).map g) :
                (a :
                    (fun (X : Type (u + 1)) ↦ X)
                      (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op j))) →
                  (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
              xj :
            (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
⊢ ∀ (j : Iᵒᵖ)
    (xi : (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op i)))
    (xj : (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj j)),
    Eq (α :=
        (fun (X : Type (u + 1)) ↦ X)
          (obj (obj (const Iᵒᵖ) ((locallyConstantPresheaf X).mapCocone c.op).pt) (Opposite.op i)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (((locallyConstantPresheaf X).mapCocone c.op).ι.app (Opposite.op i)) :
              (a :
                  (fun (X : Type (u + 1)) ↦ X)
                    (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op i))) →
                (fun (X : Type (u + 1)) ↦ X)
                  (((const Iᵒᵖ).obj ((locallyConstantPresheaf X).mapCocone c.op).pt).obj (Opposite.op i)))
            xi :
          (fun (X : Type (u + 1)) ↦ X)
            (obj (obj (const Iᵒᵖ) ((locallyConstantPresheaf X).mapCocone c.op).pt) (Opposite.op i)))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (((locallyConstantPresheaf X).mapCocone c.op).ι.app j) :
              (a : (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj j)) →
                (fun (X : Type (u + 1)) ↦ X) (((const Iᵒᵖ).obj ((locallyConstantPresheaf X).mapCocone c.op).pt).obj j))
            xj :
          (fun (X : Type (u + 1)) ↦ X) (obj (obj (const Iᵒᵖ) ((locallyConstantPresheaf X).mapCocone c.op).pt) j)) →
      ∃ (k : Iᵒᵖ) (f : Opposite.op i ⟶ k) (g : j ⟶ k),
        Eq (α := (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).map f) :
                (a :
                    (fun (X : Type (u + 1)) ↦ X)
                      (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj (Opposite.op i))) →
                  (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
              xi :
            (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).map g) :
                (a : (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj j)) →
                  (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))
              xj :
            (fun (X : Type (u + 1)) ↦ X) (((F ⋙ toProfinite).op ⋙ locallyConstantPresheaf X).obj k))

Tactic:
⟨j⟩