-- Module: Mathlib.Condensed.Discrete.LocallyConstant
Goal:
P : TopCat.{u} → Prop
inst : ∀ (S : CompHausLike P) (p : (↑(S.toTop (P := P)) : Type u) → Prop), HasProp P (Subtype p)
inst_1 : HasProp P PUnit.{u + 1}
inst_2 : HasExplicitFiniteCoproducts P
inst_3 : HasExplicitPullbacks P
hs :
  ∀ ⦃X Y : CompHausLike P⦄ (f : X ⟶ Y),
    EffectiveEpi f →
      Surjective (α := (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) X) (β :=
        (fun (X : CompHausLike P) ↦ (↑(X.toTop (P := P)) : Type u)) Y)
        (⇑(ConcreteCategory.hom (FC := fun (x1 x2 : CompHausLike P) ↦
              C((↑(x1.toTop (P := P)) : Type u), (↑(x2.toTop (P := P)) : Type u))) (CC := fun (X : CompHausLike P) ↦
              (↑(X.toTop (P := P)) : Type u)) f) :
          (a : (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) X) → (fun (X : CompHausLike P) ↦ (↑X.toTop : Type u)) Y)
inst_4 : HasExplicitFiniteCoproducts P
X : Type (max u w)
w :
  ∀ (S : CompHausLike P) (f : LocallyConstant (↑(S.toTop (P := P)) : Type u) X),
    Eq (α :=
      (ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf (coherentTopology (CompHausLike P)))
            (Functor.obj (𝟭 (Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w))))
                (Functor.obj
                  (functor.{u, w} P hs :
                    Type (max u w) ⥤ Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
                  X) :
              Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))).obj
        (op S))
      (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app
            (CategoryStruct.comp (Y :=
              Functor.obj (functorToPresheaves.{u, w} (P := P))
                ((Functor.comp (D := Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
                      (functor.{u, w} P hs :
                        Type (max u w) ⥤
                          Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
                      (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                          (op (of P PUnit.{u + 1})) :
                        Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)) ⥤
                          Type (max u w))).obj
                  X))
              (CategoryTheory.Functor.map (functorToPresheaves.{u, w} (P := P)) (NatTrans.app (unit P hs) X))
              (InducedCategory.Hom.hom
                (NatTrans.app (counit P hs)
                  (Functor.obj
                    (functor.{u, w} P hs :
                      Type (max u w) ⥤ Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
                    X))))
            (op S)))
        f)
      (TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (NatTrans.app (𝟙 (Functor.obj (functorToPresheaves.{u, w} (P := P)) X)) (op S)))
        f)
⊢ Eq (α :=
    Functor.obj (functorToPresheaves.{u, w} (P := P)) (Functor.obj (𝟭 (Type (max u w))) X) ⟶
      ObjectProperty.FullSubcategory.obj (P := Presheaf.IsSheaf (coherentTopology (CompHausLike P)))
        (Functor.obj (𝟭 (Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w))))
            (Functor.obj
              (functor.{u, w} P hs :
                Type (max u w) ⥤ Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
              X) :
          Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w))))
    (CategoryStruct.comp (Y :=
      Functor.obj (functorToPresheaves.{u, w} (P := P))
        ((Functor.comp (D := Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
              (functor.{u, w} P hs :
                Type (max u w) ⥤ Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
              (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                  (op (of P PUnit.{u + 1})) :
                Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)) ⥤ Type (max u w))).obj
          X))
      (CategoryTheory.Functor.map (functorToPresheaves.{u, w} (P := P)) (NatTrans.app (unit P hs) X))
      (InducedCategory.Hom.hom
        (NatTrans.app (counit P hs)
          (Functor.obj
            (functor.{u, w} P hs :
              Type (max u w) ⥤ Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
            X))))
    (𝟙 (Functor.obj (functorToPresheaves.{u, w} (P := P)) X))

Tactic:
ext ⟨S⟩ (f : LocallyConstant _ X)

Premises used: LocallyConstant