-- Module: Mathlib.CategoryTheory.Sites.Point.Basic
Goal:
C : Type u
inst : Category.{v, u} C
J : GrothendieckTopology.{v, u} C
Φ : Point.{w, v, u} J
A : Type u'
inst_1 : Category.{v', u'} A
inst_2 : HasColimitsOfSize.{w, w, v', u'} A
FC : A → A → Type u_1
CC : A → Type w'
inst_3 : (X Y : A) → FunLike.{u_1 + 1, w' + 1, w' + 1} (FC X Y) (CC X) (CC Y)
inst_4 : ConcreteCategory.{w', v', u', u_1} A FC (CC := CC)
P : Functor.{v, v', u, u'} Cᵒᵖ A
inst_5 : PreservesFilteredColimitsOfSize.{w, w, v', w', u', w' + 1} (forget A (FC := FC) (CC := CC))
inst_6 : LocallySmall.{w, v, u} C
X : C
x : Φ.fiber.obj X
z₁ z₂ :
  (((CategoryOfElements.π Φ.fiber).op ⋙ P) ⋙ forget A (FC := FC) (CC := CC)).obj
    (op (Sigma.mk (β := fun (c : C) ↦ Φ.fiber.obj c) X x))
⊢ ∃ (X_1 : C) (x_1 : Φ.fiber.obj X_1) (z₁_1 : ToType.{w', v', u', u_1} (FC := FC) (CC := CC) (P.obj (op X_1))) (z₂_1 :
    ToType.{w', v', u', u_1} (FC := FC) (CC := CC) (P.obj (op X_1))),
    Eq (α := CC (Functor.obj Φ.presheafFiber P))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (Φ.toPresheafFiber X_1 x_1 P) :
              (a : CC (P.obj (op X_1))) → CC (Φ.presheafFiber.obj P))
            z₁_1 :
          CC (Functor.obj Φ.presheafFiber P))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app
                  (Cocone.ι
                    (Functor.mapCocone (forget A (FC := FC) (CC := CC))
                      (colimit.cocone ((CategoryOfElements.π Φ.fiber).op ⋙ P))))
                  (op (Sigma.mk (β := fun (c : C) ↦ Φ.fiber.obj c) X x))) :
              (a : (fun (X : Type w') ↦ X) ((((CategoryOfElements.π Φ.fiber).op ⋙ P) ⋙ forget A).obj (op ⟨X, x⟩))) →
                (fun (X : Type w') ↦ X)
                  (((Functor.const Φ.fiber.Elementsᵒᵖ).obj
                        ((forget A).mapCocone (colimit.cocone ((CategoryOfElements.π Φ.fiber).op ⋙ P))).pt).obj
                    (op ⟨X, x⟩)))
            z₁ :
          (fun (X : Type w') ↦ X)
            (Functor.obj
              (Functor.obj (Functor.const Φ.fiber.Elementsᵒᵖ)
                (Cocone.pt (F := ((CategoryOfElements.π Φ.fiber).op ⋙ P) ⋙ forget A (FC := FC) (CC := CC))
                  (Functor.mapCocone (forget A (FC := FC) (CC := CC))
                    (colimit.cocone ((CategoryOfElements.π Φ.fiber).op ⋙ P)))))
              (op (Sigma.mk (β := fun (c : C) ↦ Φ.fiber.obj c) X x)))) ∧
      Eq (α := CC (Functor.obj Φ.presheafFiber P))
        ((ConcreteCategory.hom (FC := FC) (CC := CC) (Φ.toPresheafFiber X_1 x_1 P) :
              (a : CC (P.obj (op X_1))) → CC (Φ.presheafFiber.obj P))
            z₂_1 :
          CC (Functor.obj Φ.presheafFiber P))
        ((ConcreteCategory.hom (FC := TypeCat.Fun)
                (NatTrans.app
                  (Cocone.ι
                    (Functor.mapCocone (forget A (FC := FC) (CC := CC))
                      (colimit.cocone ((CategoryOfElements.π Φ.fiber).op ⋙ P))))
                  (op (Sigma.mk (β := fun (c : C) ↦ Φ.fiber.obj c) X x))) :
              (a : (fun (X : Type w') ↦ X) ((((CategoryOfElements.π Φ.fiber).op ⋙ P) ⋙ forget A).obj (op ⟨X, x⟩))) →
                (fun (X : Type w') ↦ X)
                  (((Functor.const Φ.fiber.Elementsᵒᵖ).obj
                        ((forget A).mapCocone (colimit.cocone ((CategoryOfElements.π Φ.fiber).op ⋙ P))).pt).obj
                    (op ⟨X, x⟩)))
            z₂ :
          (fun (X : Type w') ↦ X)
            (Functor.obj
              (Functor.obj (Functor.const Φ.fiber.Elementsᵒᵖ)
                (Cocone.pt (F := ((CategoryOfElements.π Φ.fiber).op ⋙ P) ⋙ forget A (FC := FC) (CC := CC))
                  (Functor.mapCocone (forget A (FC := FC) (CC := CC))
                    (colimit.cocone ((CategoryOfElements.π Φ.fiber).op ⋙ P)))))
              (op (Sigma.mk (β := fun (c : C) ↦ Φ.fiber.obj c) X x))))

Tactic:
⟨X, x, z₁, z₂, rfl, rfl⟩

Premises used: And.intro, Exists.intro, rfl