-- Module: Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits
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
j : J
U : (Opens (↑(colimit (F ⋙ forget C)) : Type v))ᵒᵖ
a :
  ∀ (U : Opens (↑(colimit (F ⋙ forget C)) : Type v)),
    Eq (α :=
      obj (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))) (op U) ⟶
        obj (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))) (op U))
      (NatTrans.app
        ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
              (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) :
            (a :
                op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf) ⟶
                  op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) →
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) ⟶
                unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)))
          (Quiver.Hom.op
            (map (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.map (𝟙 j)).c ≫
              Iso.inv (Pushforward.comp ((F ⋙ forget C).map (𝟙 j)) (colimit.ι (F ⋙ forget C) j) (F.obj j).presheaf) ≫
                Iso.hom (pushforwardEq ⋯ (F.obj j).presheaf))))
        (op U))
      (NatTrans.app
        ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
              (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) :
            (a :
                op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf) ⟶
                  op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) →
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) ⟶
                unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)))
          (𝟙 (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))))
        (op U))
⊢ Eq (α :=
    obj (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))) U ⟶
      obj (unop (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))) U)
    (NatTrans.app
      ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
            (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) :
          (a :
              op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf) ⟶
                op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) →
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) ⟶
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)))
        (Quiver.Hom.op
          (map (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.map (𝟙 j)).c ≫
            Iso.inv (Pushforward.comp ((F ⋙ forget C).map (𝟙 j)) (colimit.ι (F ⋙ forget C) j) (F.obj j).presheaf) ≫
              Iso.hom (pushforwardEq ⋯ (F.obj j).presheaf))))
      U)
    (NatTrans.app
      ((opEquiv (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))
            (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf)) :
          (a :
              op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf) ⟶
                op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) →
            unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)) ⟶
              unop (op ((pushforward C (colimit.ι (F ⋙ forget C) j)).obj (F.obj j).presheaf)))
        (𝟙 (op (obj (pushforward C (colimit.ι (F ⋙ forget C) j)) (F.obj j).presheaf))))
      U)

Tactic:
induction U with
    | op U =>
      simp [opEquiv]
      rfl

Premises used: CategoryTheory.opEquiv, Opposite.op