proposition: Proposition is_z_iso_disp_strict_twosided_disp_cat_weq : (∏ (x : C × C), isweq (F x)) × disp_functor_ff F ≃ is_z_iso_disp (D := disp_cat_of_strict_twosided_disp_cat) (@identity_z_iso cat_of_setcategory C) (disp_functor_to_two_sided_disp_functor (disp_functor_over_pair F)).
proof: Proof. use weqimplimpl. - intros HF. induction HF as [ HF_weq HF_ff ]. exact (to_is_z_iso_disp_strict_twosided_disp_cat HF_ff HF_weq). - intros HF. split. + exact (is_z_iso_disp_strict_twosided_disp_cat_to_ob_weq HF). + exact (is_z_iso_disp_strict_twosided_disp_cat_to_ff HF). - use isapropdirprod. + use impred ; intro. apply isapropisweq. + apply isaprop_disp_functor_ff. - apply isaprop_is_z_iso_disp. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.Categories.CategoryOfSetCategories UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq UniMath.CategoryTheory.DisplayedCats.Examples.DispFunctorPair UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.TransportLaws