Lemma disp_binprod_transportf (C C' : category) (D : disp_cat C) (D' : disp_cat C') (a b : C) (a' b' : C') (f g : a --> b) (f' g' : a' --> b') (x : D a) (y : D b) (x' : D' a') (y' : D' b') (ff : x -->[f] y) (ff' : x' -->[f'] y') (e : (f,,f') = (g,, g')) : transportf (@mor_disp (C ⊠ C') (disp_binprod D D') (a,,a') (b,,b') (x,,x') (y,,y')) e (ff,, ff') = transportf (mor_disp _ _ ) (maponpaths pr1 e) ff ,, transportf (mor_disp _ _ ) (maponpaths (dirprod_pr2) e) ff'. Proof. induction e. apply idpath. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.catiso. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalCategoriesTensored. Require Import UniMath.CategoryTheory.Monoidal.AlternativeDefinitions.MonoidalFunctorsTensored. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Functors. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions.