-- 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_1 :
  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_2 : True
x_1 :
  (↑((toProfinite.obj (F.obj k)).toTop (P := fun (X : TopCat.{u}) ↦ TotallyDisconnectedSpace.{u} (↑X : Type u))) :
    Type u)
x :
  (↑((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 =
    x_1
h :
  Eq (α := X)
    ((LocallyConstant.comap (TopCat.Hom.hom (c.π.app i).hom) fi :
          (a : (↑(((const I).obj c.pt).obj i).toTop : Type u)) → X)
        x :
      X)
    ((LocallyConstant.comap (TopCat.Hom.hom (c.π.app j).hom) fj :
          (a : (↑(((const I).obj c.pt).obj i).toTop : Type u)) → X)
        x :
      X)
refine_2 :
  Eq (α := X)
    ((fi : (a : (↑(toProfinite.obj (F.obj i)).toTop : Type u)) → X)
        ((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 i) :
              (a : (↑c.pt.toTop : Type u)) → (↑(toProfinite.obj (F.obj i)).toTop : Type u))
            x :
          (↑((toProfinite.obj (F.obj i)).toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
            Type u)) :
      X)
    ((fj : (a : (↑(toProfinite.obj (F.obj j)).toTop : Type u)) → X)
        ((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 ≫ toProfinite.map (F.map kj)) :
              (a : (↑c.pt.toTop : Type u)) → (↑(toProfinite.obj (F.obj j)).toTop : Type u))
            x :
          (↑((toProfinite.obj (F.obj j)).toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
            Type u)) :
      X)
refine_2₁ refine_2₂ refine_2₃ refine_2₄ :
  Eq (α := X)
    ((fi : (a : (↑(toProfinite.obj (F.obj i)).toTop : Type u)) → X)
        ((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 i) :
              (a : (↑c.pt.toTop : Type u)) → (↑(toProfinite.obj (F.obj i)).toTop : Type u))
            x :
          (↑((toProfinite.obj (F.obj i)).toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
            Type u)) :
      X)
    ((fj : (a : (↑(toProfinite.obj (F.obj j)).toTop : Type u)) → X)
        ((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 j) :
              (a : (↑c.pt.toTop : Type u)) → (↑(toProfinite.obj (F.obj j)).toTop : Type u))
            x :
          (↑((toProfinite.obj (F.obj j)).toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
            Type u)) :
      X)
⊢ Eq (α := X)
    ((fi : (a : (↑(toProfinite.obj (F.obj i)).toTop : Type u)) → X)
        ((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 ≫ toProfinite.map (F.map ki)) :
              (a : (↑c.pt.toTop : Type u)) → (↑(toProfinite.obj (F.obj i)).toTop : Type u))
            x :
          (↑((toProfinite.obj (F.obj i)).toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
            Type u)) :
      X)
    ((fj : (a : (↑(toProfinite.obj (F.obj j)).toTop : Type u)) → X)
        ((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 ≫ toProfinite.map (F.map kj)) :
              (a : (↑c.pt.toTop : Type u)) → (↑(toProfinite.obj (F.obj j)).toTop : Type u))
            x :
          (↑((toProfinite.obj (F.obj j)).toTop (P := fun (X : TopCat.{u}) ↦
                TotallyDisconnectedSpace.{u} (↑X : Type u))) :
            Type u)) :
      X)

Tactic:
rwa [dsimp% c.w, dsimp% c.w]

Premises used: CategoryTheory.Limits.Cone.w