-- Module: Mathlib.CategoryTheory.Sites.Coherent.LocallySurjective
Goal:
C : Type u_1
inst : Category.{v_1, u_1} C
inst_1 : Preregular.{v_1, u_1} C
inst_2 : FinitaryPreExtensive.{v_1, u_1} C
F G : Functor.{v_1, w, u_1, w + 1} Cᵒᵖ (Type w)
f : F ⟶ G
inst_3 : PreservesFiniteProducts F
inst_4 : PreservesFiniteProducts G
U : C
y : ToType (FC := TypeCat.Fun) (G.obj (op U))
α : Type
w : Finite α
Z : α → C
π : (a : α) → Z a ⟶ U
h : EffectiveEpiFamily Z π
h' : ∀ (a : α), (Presheaf.imageSieve (FA := TypeCat.Fun) f y).arrows (π a)
x : (a : α) → F.obj (op (Z a)) :=
  fun (a : α) ↦
    Exists.choose (p := fun (t : ToType (FC := TypeCat.Fun) (F.obj (op (Z a)))) ↦
      Eq (α := (fun (X : Type w) ↦ X) (G.obj (op (Z a))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op (Z a))) :
              (a_1 : (fun (X : Type w) ↦ X) (F.obj (op (Z a)))) → (fun (X : Type w) ↦ X) (G.obj (op (Z a))))
            t :
          (fun (X : Type w) ↦ X) (G.obj (op (Z a))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map (π a).op) :
              (a_1 : (fun (X : Type w) ↦ X) (G.obj (op U))) → (fun (X : Type w) ↦ X) (G.obj (op (Z a))))
            y :
          (fun (X : Type w) ↦ X) (G.obj (op (Z a)))))
      ⋯
x_1 : Fintype α := Fintype.ofFinite α
i' : ((a : α) → F.obj (op (Z a))) ≅ F.obj (op (sigmaObj.{0, v_1, u_1} Z)) :=
  Iso.symm (Types.productIso fun (a : α) ↦ F.obj (op (Z a))) ≪≫
    Iso.symm (PreservesProduct.iso F fun (a : α) ↦ op (Z a)) ≪≫ F.mapIso (Iso.symm (opCoproductIsoProduct Z))
this : PreservesLimitsOfShape (Discrete α)ᵒᵖ G
a :
  ∀ (j : (Discrete α)ᵒᵖ),
    Eq (α := (Functor.op (Discrete.functor Z) ⋙ G).obj j)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((G.mapCone (Cocone.op (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)))).π.app j) :
            (a : ((Functor.const (Discrete α)ᵒᵖ).obj (G.mapCone (Cocone.op (Cofan.mk (∐ Z) (Sigma.ι Z)))).pt).obj j) →
              ((Discrete.functor Z).op ⋙ G).obj j)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op (sigmaObj.{0, v_1, u_1} Z))) :
                (a : (fun (X : Type w) ↦ X) (F.obj (op (∐ Z)))) → (fun (X : Type w) ↦ X) (G.obj (op (∐ Z))))
              ((ConcreteCategory.hom (FC := TypeCat.Fun) i'.hom :
                    (a : (fun (X : Type w) ↦ X) ((a : α) → F.obj (op (Z a)))) →
                      (fun (X : Type w) ↦ X) (F.obj (op (∐ Z))))
                  x :
                (fun (X : Type w) ↦ X) (F.obj (op (sigmaObj.{0, v_1, u_1} Z)))) :
            (fun (X : Type w) ↦ X) (G.obj (op (sigmaObj.{0, v_1, u_1} Z)))) :
        (Functor.op (Discrete.functor Z) ⋙ G).obj j)
      ((ConcreteCategory.hom (FC := TypeCat.Fun)
              ((G.mapCone (Cocone.op (Cofan.mk (sigmaObj.{0, v_1, u_1} Z) (Sigma.ι Z)))).π.app j) :
            (a : ((Functor.const (Discrete α)ᵒᵖ).obj (G.mapCone (Cocone.op (Cofan.mk (∐ Z) (Sigma.ι Z)))).pt).obj j) →
              ((Discrete.functor Z).op ⋙ G).obj j)
          ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map (Quiver.Hom.op (Sigma.desc π))) :
                (a : (fun (X : Type w) ↦ X) (G.obj (op U))) → (fun (X : Type w) ↦ X) (G.obj (op (∐ Z))))
              y :
            (fun (X : Type w) ↦ X) (G.obj (op (sigmaObj.{0, v_1, u_1} Z)))) :
        (Functor.op (Discrete.functor Z) ⋙ G).obj j)
⊢ Eq (α := (fun (X : Type w) ↦ X) (G.obj (op (sigmaObj.{0, v_1, u_1} Z))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (f.app (op (sigmaObj.{0, v_1, u_1} Z))) :
          (a : (fun (X : Type w) ↦ X) (F.obj (op (∐ Z)))) → (fun (X : Type w) ↦ X) (G.obj (op (∐ Z))))
        ((ConcreteCategory.hom (FC := TypeCat.Fun) i'.hom :
              (a : (fun (X : Type w) ↦ X) ((a : α) → F.obj (op (Z a)))) → (fun (X : Type w) ↦ X) (F.obj (op (∐ Z))))
            x :
          (fun (X : Type w) ↦ X) (F.obj (op (sigmaObj.{0, v_1, u_1} Z)))) :
      (fun (X : Type w) ↦ X) (G.obj (op (sigmaObj.{0, v_1, u_1} Z))))
    ((ConcreteCategory.hom (FC := TypeCat.Fun) (G.map (Quiver.Hom.op (Sigma.desc π))) :
          (a : (fun (X : Type w) ↦ X) (G.obj (op U))) → (fun (X : Type w) ↦ X) (G.obj (op (∐ Z))))
        y :
      (fun (X : Type w) ↦ X) (G.obj (op (sigmaObj.{0, v_1, u_1} Z))))

Tactic:
apply Concrete.isLimit_ext _ (isLimitOfPreserves G (coproductIsCoproduct Z).op)

Premises used: CategoryTheory.Limits.coproductIsCoproduct, CategoryTheory.Limits.IsColimit.op, CategoryTheory.Limits.Concrete.isLimit_ext, CategoryTheory.Limits.isLimitOfPreserves