Lemma disp_functor_composite_axioms {C C' C'' : category} {D} {D'} {D''} {F : functor C C'} {F' : functor C' C''} (FF : disp_functor F D D') (FF' : disp_functor F' D' D'') : disp_functor_axioms (disp_functor_composite_data FF FF'). Proof. split; simpl. + intros x xx. etrans. apply maponpaths. apply disp_functor_id. etrans. apply disp_functor_transportf. etrans. apply maponpaths. apply disp_functor_id. etrans. apply transport_f_f. unfold transportb. apply maponpaths_2, homset_property. + intros. etrans. apply maponpaths. apply disp_functor_comp. etrans. apply disp_functor_transportf. etrans. apply maponpaths. apply disp_functor_comp. etrans. apply transport_f_f. unfold transportb. apply maponpaths_2, homset_property. Qed.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos.