Lemma transportf_precompose_disp {C} {D : disp_cat C} {c d : C} (f : c --> d ) {cc cc' : D c} (e : cc = cc') {dd} (ff : cc -->[f] dd) : transportf (λ xx : D c, xx -->[f] dd) e ff = transportf _ (id_left _) (z_iso_inv_from_z_iso_disp (idtoiso_disp (idpath _) (e)) ;; ff). Proof. destruct e; cbn. rewrite (@id_left_disp _ _ _ _ _ cc). apply pathsinv0, transportfbinv. Qed.
Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.PartA. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.DisplayedCats.Core.