-- 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)
S : CompHausLike P
Y : (CompHausLike P)ᵒᵖ ⥤ Type (max u w)
inst_1 : HasProp P PUnit.{u + 1}
f : LocallyConstant (↑(S.toTop (P := P)) : Type u) (Y.obj (op (of P PUnit.{u + 1})))
T : CompHausLike P
g : T ⟶ S
X_1 : TopCat
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 : Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w))
h :
  ∀ (x : X.obj (P := Presheaf.IsSheaf (coherentTopology (CompHausLike P))).obj (op (of P PUnit.{u + 1}))),
    TypeCat.Fun.toFun
        (ConcreteCategory.hom (FC := TypeCat.Fun)
          (CategoryStruct.comp (Y :=
            Functor.obj (C := 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))
              (Functor.obj
                  (functor.{u, w} P hs :
                    Type (max u w) ⥤ Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
                  (Functor.obj
                    (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                      (op (of P PUnit.{u + 1})))
                    X) :
                Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w))))
            (NatTrans.app (unit P hs)
                (Functor.obj
                  (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                    (op (of P PUnit.{u + 1})))
                  X) :
              Functor.obj (𝟭 (Type (max u w)))
                  (Functor.obj
                    (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                      (op (of P PUnit.{u + 1})))
                    X) ⟶
                (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
                  (Functor.obj
                    (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                      (op (of P PUnit.{u + 1})))
                    X))
            (CategoryTheory.Functor.map
              (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                (op (of P PUnit.{u + 1})))
              (NatTrans.app (counit P hs) X))))
        x =
      (ConcreteCategory.hom (FC := TypeCat.Fun)
            (𝟙
              (Functor.obj
                (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                  (op (of P PUnit.{u + 1})))
                X))).toFun
        x
⊢ CategoryStruct.comp (Y :=
      Functor.obj (C := 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))
        (Functor.obj
            (functor.{u, w} P hs :
              Type (max u w) ⥤ Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w)))
            (Functor.obj
              (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                (op (of P PUnit.{u + 1})))
              X) :
          Sheaf (C := CompHausLike P) (coherentTopology (CompHausLike P)) (Type (max u w))))
      (NatTrans.app (unit P hs)
          (Functor.obj
            (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w))) (op (of P PUnit.{u + 1})))
            X) :
        Functor.obj (𝟭 (Type (max u w)))
            (Functor.obj
              (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                (op (of P PUnit.{u + 1})))
              X) ⟶
          (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
            (Functor.obj
              (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w)))
                (op (of P PUnit.{u + 1})))
              X))
      (CategoryTheory.Functor.map
        (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w))) (op (of P PUnit.{u + 1})))
        (NatTrans.app (counit P hs) X)) =
    𝟙
      (Functor.obj
        (Functor.obj (sheafSections (coherentTopology (CompHausLike P)) (Type (max u w))) (op (of P PUnit.{u + 1}))) X)

Tactic:
ext (x : X.obj.obj _)

Premises used: CategoryTheory.ObjectProperty.FullSubcategory.obj, CategoryTheory.Functor.obj