-- Module: Mathlib.CategoryTheory.Limits.Indization.FilteredColimits
Goal:
C : Type u
inst : Category.{v, u} C
I : Type v
inst_1 : SmallCategory I
F : I ⥤ Functor.{v, v, u, v + 1} Cᵒᵖ (Type v)
J : Type v
inst_2 : SmallCategory J
inst_3 : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst_4 : SmallCategory K
H : K ⥤ Over (colimit F)
inst_5 : IsFiltered K
t : limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (colimit H))
t₂ : (fun (X : Type (max u v)) ↦ X)
  (limit
    (colimit
      (H ⋙
        yoneda ⋙
          obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
            (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))))) :=
  (ConcreteCategory.hom (FC := TypeCat.Fun) (limMap (compYonedaColimitIsoColimitCompYoneda F G H).hom) :
      (a :
          (fun (X : Type (max u v)) ↦ X)
            (limit ((G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op) ⋙ yoneda.obj (colimit H)))) →
        (fun (X : Type (max u v)) ↦ X)
          (limit
            (colimit
              (H ⋙
                yoneda ⋙
                  (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                    (G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op)))))
    t
t₃ : (fun (X : Type (max u v)) ↦ X)
  (colimit
    (limit
      (H ⋙
          yoneda ⋙
            obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
              (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))).flip)) :=
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (Iso.inv
            (colimitLimitIso
              (H ⋙
                  yoneda ⋙
                    obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
                      (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))).flip)) :
        TypeCat.Fun
          (limit
            (colimit
              (H ⋙
                    yoneda ⋙
                      obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
                        (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))).flip.flip))
          (colimit
            (limit
              (H ⋙
                  yoneda ⋙
                    obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
                      (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))).flip))) :
      (a :
          (fun (X : Type (max u v)) ↦ X)
            (limit
              (colimit
                (H ⋙
                      yoneda ⋙
                        (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                          (G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op)).flip.flip))) →
        (fun (X : Type (max u v)) ↦ X)
          (colimit
            (limit
              (H ⋙
                  yoneda ⋙
                    (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                      (G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op)).flip)))
    t₂
obtain :
  ∀ (k : K)
    (y :
      (limit
            (H ⋙
                yoneda ⋙
                  obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
                    (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))).flip).obj
        k),
    ∃ (k : K),
      Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (H.obj k)))
⊢ ∃ (k : K), Nonempty (limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (H.obj k)))

Tactic:
obtain ⟨k, y, -⟩ := Types.jointly_surjective'.{v, max u v} t₃