-- Module: Mathlib.Condensed.Discrete.Colimit
Goal:
X : LightProfiniteᵒᵖ ⥤ Type u
inst : PreservesFiniteProducts X
hX : (S : LightProfinite.{u}) → IsColimit (X.mapCocone (coconeRightOpOfCone S.asLimitCone))
S : LightProfiniteᵒᵖ
Y : FintypeCatᵒᵖ
right : Discrete PUnit.{1}
g : toLightProfinite.op.obj Y ⟶ obj (fromPUnit S) right
f :
  LocallyConstant
    (↑((toLightProfinite.obj (Opposite.unop Y)).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u)
    (X.obj
      (Opposite.op
        (CompHausLike.of
          (fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))
          PUnit.{u + 1})))
x :
  Function.Fiber (Y :=
    (↑((toLightProfinite.obj (Opposite.unop Y)).toTop (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))) :
      Type u))
    (Z :=
    X.obj
      (Opposite.op
        (CompHausLike.of
          (fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u))
          PUnit.{u + 1})))
    (⇑f :
      (a : (↑(toLightProfinite.obj (Opposite.unop Y)).toTop : Type u)) →
        X.obj
          (Opposite.op
            (CompHausLike.of
              (fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u))
              PUnit.{u + 1})))
w :
  Eq (α :=
    X.obj
      (Opposite.op
        (fiber (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (X.map
              (Quiver.Hom.op
                (sigmaIncl (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x))) :
          (a : X.obj (Opposite.op (LightProfinite.of (Opposite.unop Y).obj))) → X.obj (Opposite.op (fiber f x)))
        (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                  (Iso.inv (isoFinYonedaComponents X (LightProfinite.of (Opposite.unop Y).obj))) :
                TypeCat.Fun ((Opposite.unop Y).obj → X.obj (Opposite.op (LightProfinite.of PUnit.{u + 1})))
                  (X.obj (Opposite.op (LightProfinite.of (Opposite.unop Y).obj)))) :
              (a : (Opposite.unop Y).obj → X.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))) →
                X.obj (Opposite.op (LightProfinite.of (Opposite.unop Y).obj)))
            (⇑f : (a : (Opposite.unop Y).obj) → X.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))) :
          X.obj (Opposite.op (LightProfinite.of (Opposite.unop Y).obj))) :
      X.obj
        (Opposite.op
          (fiber (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (X.map
              (Quiver.Hom.op
                (IsTerminal.from
                  (CompHausLike.isTerminalPUnit (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)))
                  (fiber (P := fun (X : TopCat.{u}) ↦
                    TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x)))) :
          (a :
              X.obj
                (Opposite.op
                  (CompHausLike.of
                    (fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u))
                    PUnit.{u + 1}))) →
            X.obj (Opposite.op (fiber f x)))
        (Function.Fiber.image
          (⇑f :
            (a : (Opposite.unop Y).obj) →
              X.obj
                (Opposite.op
                  (CompHausLike.of
                    (fun (X : TopCat) ↦ TotallyDisconnectedSpace (↑X : Type u) ∧ SecondCountableTopology (↑X : Type u))
                    PUnit.{u + 1})))
          x) :
      X.obj
        (Opposite.op
          (fiber (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x)))
⊢ Eq (α :=
    X.obj
      (Opposite.op
        (fiber (P := fun (X : TopCat.{u}) ↦
          TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x)))
    ((ConcreteCategory.hom (FC := TypeCat.Fun)
            (X.map
              (Quiver.Hom.op
                (sigmaIncl (P := fun (X : TopCat.{u}) ↦
                  TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x))) :
          (a : X.obj (Opposite.op (LightProfinite.of (Opposite.unop Y).obj))) → X.obj (Opposite.op (fiber f x)))
        (((@ConcreteCategory.hom _ types TypeCat.Fun _ _ _ _ _
                  (Iso.inv (isoFinYonedaComponents X (LightProfinite.of (Opposite.unop Y).obj))) :
                TypeCat.Fun ((Opposite.unop Y).obj → X.obj (Opposite.op (LightProfinite.of PUnit.{u + 1})))
                  (X.obj (Opposite.op (LightProfinite.of (Opposite.unop Y).obj)))) :
              (a : (Opposite.unop Y).obj → X.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))) →
                X.obj (Opposite.op (LightProfinite.of (Opposite.unop Y).obj)))
            (⇑f : (a : (Opposite.unop Y).obj) → X.obj (Opposite.op (LightProfinite.of PUnit.{u + 1}))) :
          X.obj (Opposite.op (LightProfinite.of (Opposite.unop Y).obj))) :
      X.obj
        (Opposite.op
          (fiber (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x)))
    (counitAppAppImage (P := fun (X : TopCat.{u}) ↦
        TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x :
      X.obj
        (Opposite.op
          (fiber (P := fun (X : TopCat.{u}) ↦
            TotallyDisconnectedSpace.{u} (↑X : Type u) ∧ SecondCountableTopology.{u} (↑X : Type u)) f x)))

Tactic:
simp only [counitAppAppImage]

Premises used: CompHausLike.LocallyConstant.counitAppAppImage