proposition: Lemma disp_nat_trans_id_left {C' C : category} {D' : disp_cat C'} {D : disp_cat C} (F' F : functor C' C) (a : nat_trans F' F) (FF' : disp_functor F' D' D) (FF : disp_functor F D' D) (b : disp_nat_trans a FF' FF) : disp_nat_trans_comp (disp_nat_trans_id FF') b = transportb (λ f : nat_trans F' F, disp_nat_trans f FF' FF) (id_left (a : [C', C] ⟦ _ , _ ⟧)) b.
proof: Proof. apply subtypePath. { intro. apply isaprop_disp_nat_trans_axioms. } apply funextsec; intro c'. apply funextsec; intro xx'. apply pathsinv0. etrans. apply disp_nat_trans_transportf. apply pathsinv0. etrans. apply id_left_disp. unfold transportb. apply maponpaths_2, homset_property. Qed.
imports: UniMath.Foundations.Sets UniMath.MoreFoundations.PartA UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Isos