Definition is_disp_invertible_2cell_lwhisker {B : bicat} {x y z : B} {f : x --> y} {g₁ g₂ : y --> z} {α : invertible_2cell g₁ g₂} {D : disp_bicat B} {xx : D x} {yy : D y} {zz : D z} (ff : xx -->[f] yy) {gg₁ : yy -->[g₁] zz} {gg₂ : yy -->[g₂] zz} (αα : disp_invertible_2cell α gg₁ gg₂) : is_disp_invertible_2cell (is_invertible_2cell_lwhisker f (pr2 α)) (ff ◃◃ αα). Proof. use tpair. - exact (ff ◃◃ disp_inv_cell αα). - split. + abstract (refine (disp_lwhisker_vcomp _ _ @ _) ; refine (maponpaths _ (maponpaths _ (disp_vcomp_rinv _)) @ _) ; unfold transportb ; rewrite disp_rwhisker_transport_right ; rewrite disp_lwhisker_id2 ; unfold transportb ; rewrite !transport_f_f ; apply (transportf_paths (λ p, _ ==>[ p ] _)) ; apply B). + abstract (refine (disp_lwhisker_vcomp _ _ @ _) ; refine (maponpaths _ (maponpaths _ (disp_vcomp_linv _)) @ _) ; unfold transportb ; rewrite disp_rwhisker_transport_right ; rewrite disp_lwhisker_id2 ; unfold transportb ; rewrite !transport_f_f ; apply (transportf_paths (λ p, _ ==>[ p ] _)) ; apply B). Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.Bicategories.DisplayedBicats.DispBicat. Import DispBicat.Notations. Require Import UniMath.Bicategories.Core.Unitors. Require Import UniMath.Bicategories.Core.Univalence. Require Import UniMath.Bicategories.Core.Invertible_2cells.