proposition: Lemma precomp_tensorunit_is_ff : fully_faithful precomp_tensorunit_functor.
proof: Proof. apply disp_functor_ff_to_total_ff. { apply precomp_fully_faithful.pre_composition_with_ess_surj_and_fully_faithful_is_fully_faithful. - exact H_eso. - exact H_ff. } apply disp_prod_functor_over_fixed_base_ff. - exact (HT_ff Duniv Euniv H_eso H_ff TC TE). - exact (HU_ff I IE). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.catiso UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorCategory UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.TotalCategoryFacts UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Monoidal.RezkCompletion.LiftedTensor