Definition total_functor_comp {C₁ C₂ C₃ : category} {F : C₁ ⟶ C₂} {G : C₂ ⟶ C₃} {D₁ : disp_cat C₁} {D₂ : disp_cat C₂} {D₃ : disp_cat C₃} (FF : disp_functor F D₁ D₂) (GG : disp_functor G D₂ D₃) : total_functor FF ∙ total_functor GG ⟹ total_functor (disp_functor_composite FF GG). Proof. use make_nat_trans. - exact (λ _, identity _). - abstract (intros x y f ; refine (@id_right (total_category _) _ _ _ @ _) ; exact (!(@id_left (total_category _) _ _ _))). Defined.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.AxiomOfChoice. 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.