-- 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
c : Cocone { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C }
h : IsColimit c
mk :
  ∀ (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
        { 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 } })
    (s :
      Cocone ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C } ⋙ 𝟭 C))
    (m :
      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 } }) ⟶
        s.pt),
    (∀ (j : J),
        NatTrans.app
              (Cocone.ι
                (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) } }))
              j ≫
            m =
          s.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 } ⋙
                    𝟭 C)) ↦
            (h.desc { pt := s.pt, ι := { app := fun (j : J) ↦ s.ι.app j, naturality := ⋯ } } :
              { 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 ⟶
                Cocone.pt.{w', v₁, w, u₁} (J := J) (F :=
                  { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp })
                  { pt := s.pt, ι := { app := fun (j : J) ↦ s.ι.app j, naturality := ⋯ } }))
          s
⊢ ∀
    (s :
      Cocone ({ obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w', v₁, w, u₁} J C } ⋙ 𝟭 C))
    (m :
      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) c) ⟶
        s.pt),
    (∀ (j : J), NatTrans.app (Cocone.ι (Functor.mapCocone (𝟭 C) c)) j ≫ m = s.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 } ⋙
                    𝟭 C)) ↦
            (h.desc { pt := s.pt, ι := { app := fun (j : J) ↦ s.ι.app j, naturality := ⋯ } } :
              c.pt ⟶
                Cocone.pt.{w', v₁, w, u₁} (J := J) (F :=
                  { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp })
                  { pt := s.pt, ι := { app := fun (j : J) ↦ s.ι.app j, naturality := ⋯ } }))
          s

Tactic:
rcases c with ⟨_, _, _⟩