-- Module: Mathlib.CategoryTheory.Presentable.Type
Goal:
X : Type u
κ : Cardinal.{u}
hX : HasCardinalLT X κ
inst : Fact κ.IsRegular
J : Type u
x : SmallCategory J
h : IsCardinalFiltered J κ
F : J ⥤ Type u
c : Cocone F
hc : IsColimit c
this : IsFiltered J
refine_1 :
  ∀ (f : Cocone.pt (F := F ⋙ Functor.obj coyoneda (op X)) (Functor.mapCocone (Functor.obj coyoneda (op X)) c)),
    ∃ (i : J) (xi : (fun (X : Type u) ↦ X) ((F ⋙ Functor.obj coyoneda (op X)).obj i)),
      f =
        (hom (FC := TypeCat.Fun) (NatTrans.app (Cocone.ι (Functor.mapCocone (Functor.obj coyoneda (op X)) c)) i) :
            (a : (fun (X : Type u) ↦ X) ((F ⋙ coyoneda.obj (op X)).obj i)) →
              (fun (X : Type u) ↦ X) (((Functor.const J).obj ((coyoneda.obj (op X)).mapCocone c).pt).obj i))
          xi
refine_2 :
  ∀ (j : J) (f g : (fun (X : Type u) ↦ X) ((F ⋙ Functor.obj coyoneda (op X)).obj j))
    (h :
      Eq (α :=
        (fun (X : Type u) ↦ X)
          (Functor.obj
            (Functor.obj (Functor.const J)
              (Cocone.pt (F := F ⋙ Functor.obj coyoneda (op X)) (Functor.mapCocone (Functor.obj coyoneda (op X)) c)))
            j))
        ((hom (FC := TypeCat.Fun) (NatTrans.app (Cocone.ι (Functor.mapCocone (Functor.obj coyoneda (op X)) c)) j) :
              (a : (fun (X : Type u) ↦ X) ((F ⋙ coyoneda.obj (op X)).obj j)) →
                (fun (X : Type u) ↦ X) (((Functor.const J).obj ((coyoneda.obj (op X)).mapCocone c).pt).obj j))
            f :
          (fun (X : Type u) ↦ X)
            (Functor.obj
              (Functor.obj (Functor.const J)
                (Cocone.pt (F := F ⋙ Functor.obj coyoneda (op X)) (Functor.mapCocone (Functor.obj coyoneda (op X)) c)))
              j))
        ((hom (FC := TypeCat.Fun) (NatTrans.app (Cocone.ι (Functor.mapCocone (Functor.obj coyoneda (op X)) c)) j) :
              (a : (fun (X : Type u) ↦ X) ((F ⋙ coyoneda.obj (op X)).obj j)) →
                (fun (X : Type u) ↦ X) (((Functor.const J).obj ((coyoneda.obj (op X)).mapCocone c).pt).obj j))
            g :
          (fun (X : Type u) ↦ X)
            (Functor.obj
              (Functor.obj (Functor.const J)
                (Cocone.pt (F := F ⋙ Functor.obj coyoneda (op X)) (Functor.mapCocone (Functor.obj coyoneda (op X)) c)))
              j))),
    ∃ (k : J) (f_1 : j ⟶ k),
      Eq (α := (fun (X : Type u) ↦ X) ((F ⋙ Functor.obj coyoneda (op X)).obj k))
        ((hom (FC := TypeCat.Fun) ((F ⋙ Functor.obj coyoneda (op X)).map f_1) :
              (a : (fun (X : Type u) ↦ X) ((F ⋙ coyoneda.obj (op X)).obj j)) →
                (fun (X : Type u) ↦ X) ((F ⋙ coyoneda.obj (op X)).obj k))
            f :
          (fun (X : Type u) ↦ X) ((F ⋙ Functor.obj coyoneda (op X)).obj k))
        ((hom (FC := TypeCat.Fun) ((F ⋙ Functor.obj coyoneda (op X)).map f_1) :
              (a : (fun (X : Type u) ↦ X) ((F ⋙ coyoneda.obj (op X)).obj j)) →
                (fun (X : Type u) ↦ X) ((F ⋙ coyoneda.obj (op X)).obj k))
            g :
          (fun (X : Type u) ↦ X) ((F ⋙ Functor.obj coyoneda (op X)).obj k))
⊢ IsColimit (F := F ⋙ Functor.obj coyoneda (op X)) (Functor.mapCocone (Functor.obj coyoneda (op X)) c)

Tactic:
refine Types.FilteredColimit.isColimitOf' _ _ (fun f ↦ ?_) (fun j f g h ↦ ?_)

Premises used: CategoryTheory.Limits.Types.FilteredColimit.isColimitOf'