-- Lean 4 proof step from Mathlib.CategoryTheory.Limits.Constructions.Filtered (goal)
Goal:
C : Type u
inst : Category.{v, u} C
α : Type w
inst_1 : HasFiniteCoproducts.{v, u} C
inst_2 : HasColimitsOfShape.{w, w, v, u} (Finset (Discrete α)) C
F : Functor.{w, v, w, u} (Discrete α) C
s : Cocone F
m :
  Cocone.pt.{w, v, w, u} (J := Discrete α) (F := F)
      { pt := colimit (liftToFinsetObj F),
        ι :=
          Discrete.natTrans fun (j : Discrete α) ↦
            CategoryStruct.comp (Y := sigmaObj.{w, v, u} fun (x : ↥{j}) ↦ F.obj (↑x : Discrete α))
              (Sigma.ι.{w, v, u} (C := C) (fun (x : ↥{j}) ↦ F.obj (↑x : Discrete α))
                (Subtype.mk (p := fun (x : Discrete α) ↦ Membership.mem (γ := Finset (Discrete α)) {j} x) j ⋯))
              (colimit.ι (liftToFinsetObj F) {j}) :
        Cocone F } ⟶
    s.pt
h :
  ∀ (j : Discrete α),
    NatTrans.app
          (Cocone.ι
            { pt := colimit (liftToFinsetObj F),
              ι :=
                Discrete.natTrans fun (j : Discrete α) ↦
                  CategoryStruct.comp (Y := sigmaObj.{w, v, u} fun (x : ↥{j}) ↦ F.obj (↑x : Discrete α))
                    (Sigma.ι.{w, v, u} (C := C) (fun (x : ↥{j}) ↦ F.obj (↑x : Discrete α))
                      (Subtype.mk (p := fun (x : Discrete α) ↦ Membership.mem (γ := Finset (Discrete α)) {j} x) j ⋯))
                    (colimit.ι (liftToFinsetObj F) {j}) })
          j ≫
        m =
      s.ι.app j
t : Finset (Discrete α)
⊢ colimit.ι (liftToFinsetObj F) t ≫ m =
    CategoryStruct.comp (Y := colimit (liftToFinsetObj F)) (colimit.ι (liftToFinsetObj F) t)
      (colimit.desc (liftToFinsetObj F)
        { pt := s.pt,
          ι :=
            { app := fun (x : Finset (Discrete α)) ↦ Sigma.desc fun (x_1 : ↥x) ↦ s.ι.app (↑x_1 : Discrete α),
              naturality := ⋯ :
              NatTrans (liftToFinsetObj F) (Functor.obj (Functor.const (Finset (Discrete α))) s.pt) } })
have w : colimit.ι
      { obj := fun (s : Finset (Discrete α)) ↦ sigmaObj.{w, v, u} fun (x : ↥s) ↦ F.obj (↑x : Discrete α),
        map := fun {x Y : Finset (Discrete α)} (h : x ⟶ Y) ↦
          Sigma.desc fun (y : ↥x) ↦
            Sigma.ι.{w, v, u} (C := C) (fun (x : ↥Y) ↦ F.obj (↑x : Discrete α))
              (Subtype.mk (p := fun (x : Discrete α) ↦ x ∈ Y) (↑y : Discrete α) sorry),
        map_id := sorry, map_comp := sorry : Functor.{w, v, w, u} (Finset (Discrete α)) C }
      t ≫
    m =
  CategoryStruct.comp (Y :=
    colimit.{w, w, v, u} (J := Finset (Discrete α))
      { obj := fun (s : Finset (Discrete α)) ↦ sigmaObj.{w, v, u} fun (x : ↥s) ↦ F.obj (↑x : Discrete α),
        map := fun {x Y : Finset (Discrete α)} (h : x ⟶ Y) ↦
          Sigma.desc fun (y : ↥x) ↦
            Sigma.ι.{w, v, u} (C := C) (fun (x : ↥Y) ↦ F.obj (↑x : Discrete α))
              (Subtype.mk (p := fun (x : Discrete α) ↦ x ∈ Y) (↑y : Discrete α) sorry),
        map_id := sorry, map_comp := sorry })
    (colimit.ι
      { obj := fun (s : Finset (Discrete α)) ↦ sigmaObj.{w, v, u} fun (x : ↥s) ↦ F.obj (↑x : Discrete α),
        map := fun {x Y : Finset (Discrete α)} (h : x ⟶ Y) ↦
          Sigma.desc fun (y : ↥x) ↦
            Sigma.ι.{w, v, u} (C := C) (fun (x : ↥Y) ↦ F.obj (↑x : Discrete α))
              (Subtype.mk (p := fun (x : Discrete α) ↦ x ∈ Y) (↑y : Discrete α) sorry),
        map_id := sorry, map_comp := sorry }
      t)
    (colimit.desc
      { obj := fun (s : Finset (Discrete α)) ↦ sigmaObj.{w, v, u} fun (x : ↥s) ↦ F.obj (↑x : Discrete α),
        map := fun {x Y : Finset (Discrete α)} (h : x ⟶ Y) ↦
          Sigma.desc fun (y : ↥x) ↦
            Sigma.ι.{w, v, u} (C := C) (fun (x : ↥Y) ↦ F.obj (↑x : Discrete α))
              (Subtype.mk (p := fun (x : Discrete α) ↦ x ∈ Y) (↑y : Discrete α) sorry),
        map_id := sorry, map_comp := sorry }
      { pt := s.pt,
        ι :=
          { app := fun (x : Finset (Discrete α)) ↦ Sigma.desc fun (x_1 : ↥x) ↦ s.ι.app (↑x_1 : Discrete α),
            naturality := sorry } })
Tactic: dsimp [liftToFinsetObj]