-- 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
k : K
y_1 :
  (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
z : (fun (X : Type (max u v)) ↦ X)
  (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 (evaluation K (Type (max u v))) k)) :=
  ((ConcreteCategory.hom (FC := TypeCat.Fun)
          (Iso.hom
            (limitObjIsoLimitCompEvaluation
              (H ⋙
                  yoneda ⋙
                    obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
                      (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))).flip
              k)) :
        TypeCat.Fun
          ((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)
          (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 (evaluation K (Type (max u v))) k))) :
      (a :
          (fun (X : Type (max u v)) ↦ X)
            ((limit
                  (H ⋙
                      yoneda ⋙
                        (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                          (G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op)).flip).obj
              k)) →
        (fun (X : Type (max u v)) ↦ X)
          (limit
            ((H ⋙
                  yoneda ⋙
                    (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v))).obj
                      (G.op ⋙ (CostructuredArrow.toOver yoneda (colimit F)).op)).flip ⋙
              (evaluation K (Type (max u v))).obj k)))
    y_1
y : (H ⋙
        yoneda ⋙
          obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
            (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))).flip ⋙
    obj (evaluation K (Type (max u v))) k ≅
  (H ⋙
        yoneda ⋙
          obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
            (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F)))).obj
    k :=
  flipCompEvaluation
    (H ⋙
      yoneda ⋙
        obj (whiskeringLeft Jᵒᵖ (Over (colimit F))ᵒᵖ (Type (max u v)))
          (G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))))
    k
⊢ limit ((G.op ⋙ Functor.op (CostructuredArrow.toOver yoneda (colimit F))) ⋙ obj yoneda (H.obj k))

Tactic:
exact (lim.mapIso y).hom z

Premises used: CategoryTheory.ConcreteCategory.hom, CategoryTheory.Limits.lim, CategoryTheory.Iso.hom, DFunLike.coe, CategoryTheory.Functor.mapIso