-- Module: Mathlib.Topology.Sheaves.LocalPredicate
Goal:
X : TopCat
T : (↑X : Type u_1) → Type u_1
P : LocalPredicate T
x : (↑X : Type u_1)
w :
  ∀ (U V : OpenNhds x) (fU : (y : ↥(↑U : Opens (↑X : Type u_1))) → T (↑y : (↑X : Type u_1))),
    P.toPrelocalPredicate (T := T).pred (T := T) fU →
      ∀ (fV : (y : ↥(↑V : Opens (↑X : Type u_1))) → T (↑y : (↑X : Type u_1))),
        P.toPrelocalPredicate (T := T).pred (T := T) fV →
          Eq (α :=
              T
                (↑(Subtype.mk (p := fun (x_3 : (↑X : Type u_1)) ↦ x_3 ∈ (↑U : Opens (↑X : Type u_1))) x ⋯) :
                  (↑X : Type u_1)))
              (fU (Subtype.mk (p := fun (x_3 : (↑X : Type u_1)) ↦ x_3 ∈ (↑U : Opens (↑X : Type u_1))) x ⋯))
              (fV (Subtype.mk (p := fun (x_3 : (↑X : Type u_1)) ↦ x_3 ∈ (↑V : Opens (↑X : Type u_1))) x ⋯)) →
            ∃ (W : OpenNhds x) (iU : W ⟶ U) (iV : W ⟶ V),
              ∀ (w : ↥(↑W : Opens (↑X : Type u_1))),
                Eq (α :=
                  T (↑((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑U : Opens (↑X : Type u_1))) w) : (↑X : Type u_1)))
                  (fU ((iU : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑U : Opens (↑X : Type u_1))) w))
                  (fV ((iV : (a : ↥(↑W : Opens (↑X : Type u_1))) → ↥(↑V : Opens (↑X : Type u_1))) w))
U : (OpenNhds x)ᵒᵖ
fU : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T (↑x_1 : (↑X : Type u_1))
hU : P.toPrelocalPredicate (T := T).pred (T := T) fU
V : (OpenNhds x)ᵒᵖ
fV : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T (↑x_1 : (↑X : Type u_1))
hV : P.toPrelocalPredicate (T := T).pred (T := T) fV
h :
  Eq (α := T x)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
          (a : (subpresheafToTypes P.toPrelocalPredicate).stalk x) → T x)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (colimit.ι
                (Functor.obj
                  (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                    (OpenNhds.inclusion x).op)
                  (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))
                U) :
            (a :
                (fun (X : Type u_1) ↦ X)
                  ((((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                            (OpenNhds.inclusion x).op).obj
                        (subpresheafToTypes P.toPrelocalPredicate)).obj
                    U)) →
              (fun (X : Type u_1) ↦ X)
                (colimit
                  (((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                        (OpenNhds.inclusion x).op).obj
                    (subpresheafToTypes P.toPrelocalPredicate))))
          (Subtype.mk (p := fun (f : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj U))) → T (↑x_1 : (↑X : Type u_1))) ↦
            P.toPrelocalPredicate (T := T).pred (T := T) f) fU hU)) :
      T x)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (stalkToFiber (T := T) P x) :
          (a : (subpresheafToTypes P.toPrelocalPredicate).stalk x) → T x)
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
              (colimit.ι
                (Functor.obj
                  (Functor.obj (Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1))
                    (OpenNhds.inclusion x).op)
                  (subpresheafToTypes (T := T) (P.toPrelocalPredicate (T := T))))
                V) :
            (a :
                (fun (X : Type u_1) ↦ X)
                  ((((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                            (OpenNhds.inclusion x).op).obj
                        (subpresheafToTypes P.toPrelocalPredicate)).obj
                    V)) →
              (fun (X : Type u_1) ↦ X)
                (colimit
                  (((Functor.whiskeringLeft (OpenNhds x)ᵒᵖ (Opens (↑X : Type u_1))ᵒᵖ (Type u_1)).obj
                        (OpenNhds.inclusion x).op).obj
                    (subpresheafToTypes P.toPrelocalPredicate))))
          (Subtype.mk (p := fun (f : (x_1 : ↥(unop ((OpenNhds.inclusion x).op.obj V))) → T (↑x_1 : (↑X : Type u_1))) ↦
            P.toPrelocalPredicate (T := T).pred (T := T) f) fV hV)) :
      T x)
⊢ Eq (α := T x)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((P.cocone (T := T) x).ι.app U) :
          (a : { f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop U))) → T (↑x_1 : (↑X : Type u_1)) // P.pred f }) →
            (P.cocone x).pt)
        (Subtype.mk (p := fun (f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop U))) → T (↑x_1 : (↑X : Type u_1))) ↦
          P.toPrelocalPredicate (T := T).pred (T := T) f) fU hU) :
      (P.cocone (T := T) x).pt)
    ((ConcreteCategory.hom (FC := TypeCat.Fun) ((P.cocone (T := T) x).ι.app V) :
          (a : { f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop V))) → T (↑x_1 : (↑X : Type u_1)) // P.pred f }) →
            (P.cocone x).pt)
        (Subtype.mk (p := fun (f : (x_1 : ↥((OpenNhds.inclusion x).obj (unop V))) → T (↑x_1 : (↑X : Type u_1))) ↦
          P.toPrelocalPredicate (T := T).pred (T := T) f) fV hV) :
      (P.cocone (T := T) x).pt)

Tactic:
simp only [Functor.whiskeringLeft_obj_obj, Functor.comp_obj, Functor.op_obj,
    subpresheafToTypes_obj, stalkToFiber_ι] at h

Premises used: CategoryTheory.Functor.comp_obj, TopCat.stalkToFiber_ι, CategoryTheory.Functor.whiskeringLeft_obj_obj, TopCat.subpresheafToTypes_obj, CategoryTheory.Functor.op_obj