-- 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) } })
j : J
pt_1 : C
ι :
  { 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
⊢ { 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 ≫
      (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 := ι } =
    { pt := pt_1, ι := ι :
            Cocone
              { obj := obj, map := @map, map_id := map_id, map_comp := @map_comp : Functor.{w, v₁, w', u₁} J C } }.ι.app
      j

Tactic:
h.fac _ j

Premises used: CategoryTheory.Limits.IsColimit.fac