{P : Sort u_1} →
  {J : Type u} →
    {U : J → TopCat} →
      {V : (i : J) → J → TopologicalSpace.Opens ↑(U i)} →
        {t :
            (i j : J) →
              (TopologicalSpace.Opens.toTopCat (U i)).obj (V i j) ⟶
                (TopologicalSpace.Opens.toTopCat (U j)).obj (V j i)} →
          {V_id : ∀ (i : J), V i i = ⊤} →
            {t_id : ∀ (i : J), ⇑(CategoryTheory.ConcreteCategory.hom (t i i)) = id} →
              {t_inter :
                  ∀ ⦃i j : J⦄ (k : J) (x : ↥(V i j)),
                    ↑x ∈ V i k → ↑((CategoryTheory.ConcreteCategory.hom (t i j)) x) ∈ V j k} →
                {cocycle :
                    ∀ (i j k : J) (x : ↥(V i j)) (h : ↑x ∈ V i k),
                      ↑((CategoryTheory.ConcreteCategory.hom (t j k))
                            ⟨↑((CategoryTheory.ConcreteCategory.hom (t i j)) x), ⋯⟩) =
                        ↑((CategoryTheory.ConcreteCategory.hom (t i k)) ⟨↑x, h⟩)} →
                  {J' : Type u} →
                    {U' : J' → TopCat} →
                      {V' : (i : J') → J' → TopologicalSpace.Opens ↑(U' i)} →
                        {t' :
                            (i j : J') →
                              (TopologicalSpace.Opens.toTopCat (U' i)).obj (V' i j) ⟶
                                (TopologicalSpace.Opens.toTopCat (U' j)).obj (V' j i)} →
                          {V_id' : ∀ (i : J'), V' i i = ⊤} →
                            {t_id' : ∀ (i : J'), ⇑(CategoryTheory.ConcreteCategory.hom (t' i i)) = id} →
                              {t_inter' :
                                  ∀ ⦃i j : J'⦄ (k : J') (x : ↥(V' i j)),
                                    ↑x ∈ V' i k → ↑((CategoryTheory.ConcreteCategory.hom (t' i j)) x) ∈ V' j k} →
                                {cocycle' :
                                    ∀ (i j k : J') (x : ↥(V' i j)) (h : ↑x ∈ V' i k),
                                      ↑((CategoryTheory.ConcreteCategory.hom (t' j k))
                                            ⟨↑((CategoryTheory.ConcreteCategory.hom (t' i j)) x), ⋯⟩) =
                                        ↑((CategoryTheory.ConcreteCategory.hom (t' i k)) ⟨↑x, h⟩)} →
                                  { J := J, U := U, V := V, t := t, V_id := V_id, t_id := t_id, t_inter := t_inter,
                                        cocycle := cocycle } =
                                      { J := J', U := U', V := V', t := t', V_id := V_id', t_id := t_id',
                                        t_inter := t_inter', cocycle := cocycle' } →
                                    (J = J' → U ≍ U' → V ≍ V' → t ≍ t' → P) → P