proposition: Lemma functor_tensor_disp_cat_is_univalent : is_univalent_disp (functor_tensor_disp_cat TC TD).
proof: Proof. apply is_univalent_disp_from_fibers. intros F pt1 pt2. use isweqimplimpl. - intro i. use total2_paths_f. + apply funextsec ; intro. set (ix := (pr1 i) (pr1 x) (pr2 x)). cbn in ix. rewrite binprod_id in ix. rewrite (functor_id TD) in ix. rewrite id_right in ix. rewrite id_left in ix. exact (! ix). + do 2 (apply funextsec ; intro). repeat (apply impred_isaprop ; intro). apply D. - apply isaset_functor_tensor_disp_cat. - apply isaproptotal2. + intro ; apply Isos.isaprop_is_z_iso_disp. + do 4 intro ; apply isaprop_is_nat_trans_tensor. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored