Definition disp_cat_of_fibers_comp {B C : category} (F : functor C B) : ∏ (b b' b'' : B) (f : b --> b') (f': b' --> b'') (c : disp_cat_of_fibers_ob F b) (c' : disp_cat_of_fibers_ob F b') (c'' : disp_cat_of_fibers_ob F b''), (disp_cat_of_fibers_mor' c c' f) -> (disp_cat_of_fibers_mor' c' c'' f') -> (disp_cat_of_fibers_mor' c c'' (f · f')). Proof. intros b b' b'' f f' [c H_ob] [c' H'_ob] [c'' H''_ob] [g H_mor] [g' H'_mor]. exists (g · g'). eapply pathscomp0. - apply functor_comp. - eapply pathscomp0. 2: { apply pathsinv0. apply (transp_pres_comp' H_ob H'_ob H''_ob). } + eapply pathscomp0. * exact (maponpaths (λ mor, mor · ((# F) g')) H_mor). * exact (maponpaths (λ mor, (transportf_mor H_ob H'_ob f) · mor) H'_mor). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.Categories.HSET.MonoEpiIso. Require Import UniMath.CategoryTheory.Categories.HSET.Univalence. Require Import UniMath.CategoryTheory.Limits.Pullbacks. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.Presheaf. Require Import UniMath.CategoryTheory.Core.TransportMorphisms. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Fibrations. Require Import UniMath.Foundations.All. Require Import UniMath.CategoryTheory.DisplayedCats.MoreFibrations.Prefibrations. Require Import UniMath.CategoryTheory.DisplayedCats.MoreFibrations.CartesiannessOfComposites.