-- 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
    { 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 } }
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 } ⋙ 𝟭 C
⊢ CategoryStruct.comp (Y :=
      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 } } }))
      ((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 } } ⟶
                { 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))
          { pt := pt_1, π := π } :
        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 :=
                { 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 } ⋙
                        𝟭 C) }.pt,
              π :=
                {
                  app := fun (j : 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 } ⋙
                                𝟭 C) }.π.app
                      j,
                  naturality := ⋯ :
                  NatTrans
                    (Functor.obj (Functor.const 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 } ⋙
                              𝟭 C) }.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)
      (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 :
        Functor.obj.{w', v₁, w, u₁}
            (Functor.obj (Functor.const J)
              (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 } } :
                    Cone
                      { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                        Functor.{w', v₁, w, u₁} J C } })))
            j ⟶
          ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C } ⋙ 𝟭 C).obj
            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 } ⋙
                𝟭 C) }.π.app
      j

Tactic:
exact h.fac _ j

Premises used: CategoryTheory.Limits.IsLimit.fac