-- 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) →
    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }.obj X ⟶
      Functor.obj.{w, v₁, w', u₁} (Functor.obj (Functor.const J) pt) X
naturality :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y),
    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }.map f ≫ app Y =
      app X ≫ Functor.map (Functor.obj (Functor.const J) pt) f
h :
  IsColimit (F :=
    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } ⋙ 𝟭 C)
    (Functor.mapCocone (𝟭 C)
      { pt := pt,
        ι :=
          { app := app, naturality := naturality :
            NatTrans { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }
              (Functor.obj (Functor.const J) pt) } })
pt_1 : C
app_1 :
  (X : J) →
    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }.obj X ⟶
      Functor.obj.{w, v₁, w', u₁} (Functor.obj (Functor.const J) pt_1) X
naturality_1 :
  ∀ ⦃X Y : J⦄ (f : X ⟶ Y),
    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }.map f ≫ app_1 Y =
      app_1 X ≫ Functor.map (Functor.obj (Functor.const J) pt_1) f
m :
  { pt := pt,
        ι :=
          { app := app, naturality := naturality :
            NatTrans { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }
              (Functor.obj (Functor.const J) pt) } :
        Cocone { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } }.pt ⟶
    { pt := pt_1,
        ι :=
          { app := app_1, naturality := naturality_1 :
            NatTrans { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }
              (Functor.obj (Functor.const J) pt_1) } :
        Cocone { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } }.pt
w :
  ∀ (j : J),
    { pt := pt,
                ι :=
                  { app := app, naturality := naturality :
                    NatTrans
                      { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }
                      (Functor.obj (Functor.const J) pt) } :
                Cocone
                  { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                    Functor.{w, v₁, w', u₁} J C } }.ι.app
          j ≫
        m =
      { pt := pt_1,
              ι :=
                { app := app_1, naturality := naturality_1 :
                  NatTrans
                    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }
                    (Functor.obj (Functor.const J) pt_1) } }.ι.app
        j
⊢ m =
    (fun
          (s :
            Cocone { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }) ↦
        (h.desc { pt := s.pt, ι := { app := fun (j : J) ↦ s.ι.app j, naturality := ⋯ } } :
          Cocone.pt (F :=
              { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } ⋙ 𝟭 C)
              (Functor.mapCocone (𝟭 C)
                { pt := pt,
                  ι :=
                    { app := app, naturality := naturality :
                      NatTrans
                        { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                          Functor.{w, v₁, w', u₁} J C }
                        (Functor.obj (Functor.const J) pt) } :
                  Cocone
                    { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp :
                      Functor.{w, v₁, w', u₁} J C } }) ⟶
            Cocone.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, ι := { app := fun (j : J) ↦ s.ι.app j, naturality := ⋯ } }))
      { pt := pt_1,
        ι :=
          { app := app_1, naturality := naturality_1 :
            NatTrans { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C }
              (Functor.obj (Functor.const J) pt_1) } }

Tactic:
exact h.uniq _ m w

Premises used: CategoryTheory.Limits.IsColimit.uniq