-- Module: Mathlib.CategoryTheory.Limits.Preserves.Basic
Goal:
C : Type u₁
inst : Category.{v₁, u₁} C
D : Type u₂
inst_1 : Category.{v₂, u₂} D
J_1 : Type w
inst_2 : Category.{w', w} J_1
K : Functor.{w', v₁, w, u₁} J_1 C
J : Type w'
𝒥 : Category.{w, w'} J
obj : J → C
map : {X Y : J} → (X ⟶ Y) → (obj X ⟶ obj Y)
map_id : ∀ (X : J), map (𝟙 X) = 𝟙 (obj X)
map_comp : ∀ {X Y Z : J} (f : X ⟶ Y) (g : Y ⟶ Z), map (f ≫ g) = map f ≫ map g
pt : C
app :
  (X : J) →
    Functor.obj.{w, v₁, w', u₁} (Functor.obj (Functor.const J) pt) X ⟶
      { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }.obj X
naturality :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y),
    Functor.map (Functor.obj (Functor.const J) pt) f ≫ app Y =
      app X ≫ { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp }.map f
h :
  IsLimit (F :=
    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } ⋙ 𝟭 C)
    (Functor.mapCone (𝟭 C)
      { pt := pt,
        π :=
          { app := app, naturality := naturality :
            NatTrans (Functor.obj (Functor.const J) pt)
              { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } } })
j : J
pt_1 : C
π :
  Functor.obj (Functor.const J) pt_1 ⟶
    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }
⊢ (fun (s : Cone { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }) ↦
          (h.lift { pt := s.pt, π := { app := fun (j : J) ↦ s.π.app j, naturality := ⋯ } } :
            Cone.pt.{w, v₁, w', u₁} (J := J) (F :=
                { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } ⋙
                  𝟭 C)
                { pt := s.pt,
                  π :=
                    { app := fun (j : J) ↦ s.π.app j, naturality := ⋯ :
                      NatTrans (Functor.obj (Functor.const J) s.pt)
                        ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                            Functor.{w, v₁, w', u₁} J C } ⋙
                          𝟭 C) } :
                  Cone
                    ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                        Functor.{w, v₁, w', u₁} J C } ⋙
                      𝟭 C) } ⟶
              Cone.pt (F :=
                { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } ⋙
                  𝟭 C)
                (Functor.mapCone (𝟭 C)
                  { pt := pt,
                    π :=
                      { app := app, naturality := naturality :
                        NatTrans (Functor.obj (Functor.const J) pt)
                          { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                            Functor.{w, v₁, w', u₁} J C } } })))
        { pt := pt_1, π := π } ≫
      { pt := pt,
              π :=
                { app := app, naturality := naturality :
                  NatTrans (Functor.obj (Functor.const J) pt)
                    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                      Functor.{w, v₁, w', u₁} J C } } :
              Cone
                { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                  Functor.{w, v₁, w', u₁} J C } }.π.app
        j =
    { pt := pt_1, π := π :
            Cone
              { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } }.π.app
      j

Tactic:
exact h.fac _ j

Premises used: CategoryTheory.Limits.IsLimit.fac