-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Preserves.Basic (goal)
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
c : Cone { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C }
h : IsLimit c
⊢ ∀
    (s :
      Cone ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C } ⋙ 𝟭 C))
    (m :
      s.pt ⟶
        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) c)),
    (∀ (j : J), m ≫ NatTrans.app (Cone.π (Functor.mapCone (𝟭 C) c)) j = s.π.app j) →
      m =
        (fun
              (s :
                Cone
                  ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C } ⋙
                    𝟭 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 })
                  { 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 } } :
                    Cone
                      { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                        Functor.{w', v₁, w, u₁} J C } } ⟶
                c.pt))
          s
have mk : ∀ (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
      { 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 } })
  (s : Cone ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C } ⋙ 𝟭 C))
  (m :
    s.pt ⟶
      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 } } })),
  (∀ (j : J),
      m ≫
          NatTrans.app
            (Cone.π
              (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 =
        s.π.app j) →
    m =
      (fun
            (s :
              Cone
                ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C } ⋙
                  𝟭 C)) ↦
          (h.lift { pt := s.pt, π := { app := fun (j : J) ↦ s.π.app j, naturality := sorry } } :
            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 })
                { pt := s.pt,
                  π :=
                    { app := fun (j : J) ↦ s.π.app j, naturality := sorry :
                      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 } } :
                  Cone
                    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                      Functor.{w', v₁, w, u₁} J 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 } } :
                  Cone
                    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                      Functor.{w', v₁, w, u₁} J C } }.pt))
        s
Tactic: rcases c with ⟨_, _, _⟩
Cleared hypotheses: c, h