-- Module: Mathlib.Condensed.Discrete.Colimit
Goal:
F : LightProfiniteᵒᵖ ⥤ Type u
inst : PreservesFiniteProducts F
X Y : LightProfinite
inst_1 :
  Finite.{u + 1}
    (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u)
inst_2 :
  Finite.{u + 1}
    (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u)
f :
  (↑(Y.toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u) →
    F.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))
g : X ⟶ Y
a :
  ∀
    (x :
      (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
        Type u)),
    (f ∘
          (⇑(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))
                g) :
            (a : (↑X.toTop : Type u)) → (↑Y.toTop : Type u)))
        x =
      (ConcreteCategory.hom (FC := TypeCat.Fun) (isoFinYonedaComponents F X).hom :
          (a : F.obj (Opposite.op X)) → (↑X.toTop : Type u) → F.obj (Opposite.op (LightProfinite.of PUnit.{u + 1})))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g.op) : (a : F.obj (Opposite.op Y)) → F.obj (Opposite.op X))
            ((ConcreteCategory.hom (FC := TypeCat.Fun) (isoFinYonedaComponents F Y).inv :
                  (a : (↑Y.toTop : Type u) → F.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))) →
                    F.obj (Opposite.op Y))
                f :
              F.obj (Opposite.op Y)) :
          F.obj (Opposite.op X))
        x
⊢ Eq (α :=
    (↑(X.toTop (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
        Type u) →
      F.obj (Opposite.op (LightProfinite.of PUnit.{u + 1})))
    (f ∘
      (⇑(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))
            g) :
        (a : (↑X.toTop : Type u)) → (↑Y.toTop : Type u)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (isoFinYonedaComponents F X).hom :
        (a : F.obj (Opposite.op X)) → (↑X.toTop : Type u) → F.obj (Opposite.op (LightProfinite.of PUnit.{u + 1})))
      ((ConcreteCategory.hom (FC := TypeCat.Fun) (F.map g.op) : (a : F.obj (Opposite.op Y)) → F.obj (Opposite.op X))
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (isoFinYonedaComponents F Y).inv :
                (a : (↑Y.toTop : Type u) → F.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))) →
                  F.obj (Opposite.op Y))
              f :
            F.obj (Opposite.op Y)) :
        F.obj (Opposite.op X)))

Tactic:
ext x