-- Lean 4 proof step from Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits (goal)
Goal:
J : Type u'
inst : Category.{v', u'} J
C : Type u
inst_1 : Category.{v, u} C
inst_2 : HasColimitsOfShape.{v', u', v, v + 1} J TopCat
F : J ⥤ PresheafedSpace.{v, u, v} C
j₁ j₂ j₃ : J
f : j₁ ⟶ j₂
g : j₂ ⟶ j₃
⊢ Eq (α :=
    Quiver.Hom (V := Presheaf.{v, v, u} C (colimit (F ⋙ forget C)))
      (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.obj j₃).presheaf)))
      (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₁)) (F.obj j₁).presheaf))))
    ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₁)) (F.obj j₁).presheaf))
            (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.obj j₃).presheaf)) :
          (a :
              op ((pushforward C (colimit.ι (F ⋙ forget C) j₁)).obj (F.obj j₁).presheaf) ⟶
                op ((pushforward C (colimit.ι (F ⋙ forget C) j₃)).obj (F.obj j₃).presheaf)) →
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j₃)).obj (F.obj j₃).presheaf)) ⟶
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j₁)).obj (F.obj j₁).presheaf)))
        (Quiver.Hom.op
          (map (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.map (f ≫ g)).c ≫
            Iso.inv (Pushforward.comp ((F ⋙ forget C).map (f ≫ g)) (colimit.ι (F ⋙ forget C) j₃) (F.obj j₁).presheaf) ≫
              Iso.hom (pushforwardEq ⋯ (F.obj j₁).presheaf))) :
      Quiver.Hom (V := Presheaf.{v, v, u} C (colimit (F ⋙ forget C)))
        (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.obj j₃).presheaf)))
        (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₁)) (F.obj j₁).presheaf))))
    ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₁)) (F.obj j₁).presheaf))
            (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.obj j₃).presheaf)) :
          (a :
              op ((pushforward C (colimit.ι (F ⋙ forget C) j₁)).obj (F.obj j₁).presheaf) ⟶
                op ((pushforward C (colimit.ι (F ⋙ forget C) j₃)).obj (F.obj j₃).presheaf)) →
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j₃)).obj (F.obj j₃).presheaf)) ⟶
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j₁)).obj (F.obj j₁).presheaf)))
        (Quiver.Hom.op
            (map (pushforward C (colimit.ι (F ⋙ forget C) j₂)) (F.map f).c ≫
              Iso.inv (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j₂) (F.obj j₁).presheaf) ≫
                Iso.hom (pushforwardEq ⋯ (F.obj j₁).presheaf)) ≫
          Quiver.Hom.op
            (map (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.map g).c ≫
              Iso.inv (Pushforward.comp ((F ⋙ forget C).map g) (colimit.ι (F ⋙ forget C) j₃) (F.obj j₂).presheaf) ≫
                Iso.hom (pushforwardEq ⋯ (F.obj j₂).presheaf))) :
      Quiver.Hom (V := Presheaf.{v, v, u} C (colimit (F ⋙ forget C)))
        (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.obj j₃).presheaf)))
        (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₁)) (F.obj j₁).presheaf))))
have a : ∀ (U : (Opens (↑(colimit (F ⋙ forget C)) : Type v))ᵒᵖ),
  Eq (α :=
    obj (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.obj j₃).presheaf))) U ⟶
      obj (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₁)) (F.obj j₁).presheaf))) U)
    (NatTrans.app
      ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₁)) (F.obj j₁).presheaf))
            (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.obj j₃).presheaf)) :
          (a :
              op ((pushforward C (colimit.ι (F ⋙ forget C) j₁)).obj (F.obj j₁).presheaf) ⟶
                op ((pushforward C (colimit.ι (F ⋙ forget C) j₃)).obj (F.obj j₃).presheaf)) →
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j₃)).obj (F.obj j₃).presheaf)) ⟶
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j₁)).obj (F.obj j₁).presheaf)))
        (Quiver.Hom.op
          (map (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.map (f ≫ g)).c ≫
            Iso.inv (Pushforward.comp ((F ⋙ forget C).map (f ≫ g)) (colimit.ι (F ⋙ forget C) j₃) (F.obj j₁).presheaf) ≫
              Iso.hom (pushforwardEq sorry (F.obj j₁).presheaf))))
      U)
    (NatTrans.app
      ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₁)) (F.obj j₁).presheaf))
            (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.obj j₃).presheaf)) :
          (a :
              op ((pushforward C (colimit.ι (F ⋙ forget C) j₁)).obj (F.obj j₁).presheaf) ⟶
                op ((pushforward C (colimit.ι (F ⋙ forget C) j₃)).obj (F.obj j₃).presheaf)) →
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j₃)).obj (F.obj j₃).presheaf)) ⟶
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j₁)).obj (F.obj j₁).presheaf)))
        (Quiver.Hom.op
            (map (pushforward C (colimit.ι (F ⋙ forget C) j₂)) (F.map f).c ≫
              Iso.inv (Pushforward.comp ((F ⋙ forget C).map f) (colimit.ι (F ⋙ forget C) j₂) (F.obj j₁).presheaf) ≫
                Iso.hom (pushforwardEq sorry (F.obj j₁).presheaf)) ≫
          Quiver.Hom.op
            (map (pushforward C (colimit.ι (F ⋙ forget C) j₃)) (F.map g).c ≫
              Iso.inv (Pushforward.comp ((F ⋙ forget C).map g) (colimit.ι (F ⋙ forget C) j₃) (F.obj j₂).presheaf) ≫
                Iso.hom (pushforwardEq sorry (F.obj j₂).presheaf))))
      U)
Tactic: refine NatTrans.ext (funext fun U => ?_)