Lemma fiber_is_precategory : is_precategory fiber_category_data. Proof. apply is_precategory_one_assoc_to_two. repeat split; intros; cbn. - etrans. apply maponpaths. apply id_left_disp. etrans. apply transport_f_f. apply transportf_comp_lemma_hset. apply (homset_property). apply idpath. - etrans. apply maponpaths. apply id_right_disp. etrans. apply transport_f_f. apply transportf_comp_lemma_hset. apply (homset_property). apply idpath. - etrans. apply maponpaths. apply mor_disp_transportf_prewhisker. etrans. apply transport_f_f. etrans. apply maponpaths. apply assoc_disp. etrans. apply transport_f_f. apply pathsinv0. etrans. apply maponpaths. apply mor_disp_transportf_postwhisker. etrans. apply transport_f_f. apply maponpaths_2. apply homset_property. Qed.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.NaturalTransformations. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Total.