proposition: Lemma precomp_tensorunit_is_eso : essentially_surjective precomp_tensorunit_functor.
proof: Proof. apply disp_functor_eso_to_total_eso. { apply precomp_ess_surj.pre_composition_essentially_surjective. - exact Euniv. - exact H_eso. - exact H_ff. } apply disp_prod_functor_over_fixed_base_eso. - exact (HT_eso Duniv Euniv H_eso H_ff TC TE). - exact (HU_eso I IE). - use Fibrations.iso_cleaving_category. apply is_univalent_functor_category. exact Euniv. 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