-- Mathlib/CategoryTheory/Limits/Indization/FilteredColimits.lean
theorem exists_nonempty_limit_obj_of_colimit [IsFiltered K]
    (h : Nonempty <| limit <| 𝒢 ⋙ yoneda.obj (colimit H)) :
    ∃ k, Nonempty <| limit <| 𝒢 ⋙ yoneda.obj (H.obj k)

Proof state:
case intro.intro.intro
C : Type u
inst✝⁵ : Category.{v, u} C
I : Type v
inst✝⁴ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
J : Type v
inst✝³ : SmallCategory J
inst✝² : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst✝¹ : SmallCategory K
H : K ⥤ Over (colimit F)
inst✝ : IsFiltered K
t : limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (colimit H))
t₂ : limit
  (colimit
    (H ⋙
      yoneda ⋙
        (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj (G.op ⋙ (toOver yoneda (colimit F)).op))) :=
  limMap (compYonedaColimitIsoColimitCompYoneda F G H).hom t
t₃ : colimit
  (limit
    (H ⋙
        yoneda ⋙
          (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
            (G.op ⋙ (toOver yoneda (colimit F)).op)).flip) :=
  (colimitLimitIso
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip).inv
    t₂
k : K
y :
  (limit
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip).obj
    k
⊢ ∃ k, Nonempty (limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (H.obj k)))

Tactic:
refine ⟨k, ⟨?_⟩⟩

Result:
case intro.intro.intro
C : Type u
inst✝⁵ : Category.{v, u} C
I : Type v
inst✝⁴ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
J : Type v
inst✝³ : SmallCategory J
inst✝² : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst✝¹ : SmallCategory K
H : K ⥤ Over (colimit F)
inst✝ : IsFiltered K
t : limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (colimit H))
t₂ : limit
  (colimit
    (H ⋙
      yoneda ⋙
        (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj (G.op ⋙ (toOver yoneda (colimit F)).op))) :=
  limMap (compYonedaColimitIsoColimitCompYoneda F G H).hom t
t₃ : colimit
  (limit
    (H ⋙
        yoneda ⋙
          (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
            (G.op ⋙ (toOver yoneda (colimit F)).op)).flip) :=
  (colimitLimitIso
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip).inv
    t₂
k : K
y :
  (limit
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip).obj
    k
⊢ limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (H.obj k))

Proof state:
case intro.intro.intro
C : Type u
inst✝⁵ : Category.{v, u} C
I : Type v
inst✝⁴ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
J : Type v
inst✝³ : SmallCategory J
inst✝² : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst✝¹ : SmallCategory K
H : K ⥤ Over (colimit F)
inst✝ : IsFiltered K
t : limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (colimit H))
t₂ : limit
  (colimit
    (H ⋙
      yoneda ⋙
        (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj (G.op ⋙ (toOver yoneda (colimit F)).op))) :=
  limMap (compYonedaColimitIsoColimitCompYoneda F G H).hom t
t₃ : colimit
  (limit
    (H ⋙
        yoneda ⋙
          (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
            (G.op ⋙ (toOver yoneda (colimit F)).op)).flip) :=
  (colimitLimitIso
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip).inv
    t₂
k : K
y :
  (limit
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip).obj
    k
⊢ limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (H.obj k))

Tactic:
let z := (limitObjIsoLimitCompEvaluation (H ⋙ yoneda ⋙ (whiskeringLeft _ _ _).obj 𝒢).flip k).hom y

Result:
case intro.intro.intro
C : Type u
inst✝⁵ : Category.{v, u} C
I : Type v
inst✝⁴ : SmallCategory I
F : I ⥤ Cᵒᵖ ⥤ Type v
J : Type v
inst✝³ : SmallCategory J
inst✝² : FinCategory J
G : J ⥤ CostructuredArrow yoneda (colimit F)
K : Type v
inst✝¹ : SmallCategory K
H : K ⥤ Over (colimit F)
inst✝ : IsFiltered K
t : limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (colimit H))
t₂ : limit
  (colimit
    (H ⋙
      yoneda ⋙
        (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj (G.op ⋙ (toOver yoneda (colimit F)).op))) :=
  limMap (compYonedaColimitIsoColimitCompYoneda F G H).hom t
t₃ : colimit
  (limit
    (H ⋙
        yoneda ⋙
          (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
            (G.op ⋙ (toOver yoneda (colimit F)).op)).flip) :=
  (colimitLimitIso
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip).inv
    t₂
k : K
y :
  (limit
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip).obj
    k
z : limit
  ((H ⋙
        yoneda ⋙
          (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj (G.op ⋙ (toOver yoneda (colimit F)).op)).flip ⋙
    (evaluation K (Type (max u v))).obj k) :=
  (limitObjIsoLimitCompEvaluation
        (H ⋙
            yoneda ⋙
              (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                (G.op ⋙ (toOver yoneda (colimit F)).op)).flip
        k).hom
    y
⊢ limit ((G.op ⋙ (toOver yoneda (colimit F)).op) ⋙ yoneda.obj (H.obj k))