Definition total_nat_trans {C₁ C₂ : category} {F G : C₁ ⟶ C₂} {τ : F ⟹ G} {D₁ : disp_cat C₁} {D₂ : disp_cat C₂} {FF : disp_functor F D₁ D₂} {GG : disp_functor G D₁ D₂} (ττ : disp_nat_trans τ FF GG) : nat_trans (total_functor FF) (total_functor GG). Proof. use make_nat_trans. - exact (λ x, τ (pr1 x) ,, ττ (pr1 x) (pr2 x)). - abstract (intros x y f ; cbn ; use total2_paths_b ; [ exact (nat_trans_ax τ _ _ (pr1 f)) | exact (disp_nat_trans_ax ττ (pr2 f))]). 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.