proposition: Lemma two_sided_disp_cat_iso_path_mor {x₁ x₂ y₁ y₂ : C} (xx := (x₁ ,, y₁) : category_binproduct C C) (yy := (x₂ ,, y₂) : category_binproduct C C) {f : x₁ --> x₂} {g : y₁ --> y₂} {xy₁ : D₁ x₁ y₁} {xy₂ : D₁ x₂ y₂} (fg : xy₁ -->[ f ][ g ] xy₂) (fg' := fg : @mor_disp _ (twosided_disp_cat_to_disp_cat C C D₁) xx yy xy₁ xy₂ (f ,, g)) : #2 G (♯ F fg')%mor_disp = transportf_disp_mor2 (id_right _ @ id_left _) (id_right _ @ id_left _) (idtoiso_twosided_disp (idpath _) (idpath _) (two_sided_disp_cat_iso_path_ob xy₁) ;;2 fg ;;2 idtoiso_twosided_disp (idpath _) (idpath _) (!(two_sided_disp_cat_iso_path_ob xy₂))).
proof: Proof. etrans. { apply (path_twosided_disp_functor_mor (inv_mor_after_z_iso_disp HF)). } cbn -[idtoiso_twosided_disp]. etrans. { apply maponpaths. apply maponpaths_2. apply maponpaths. apply (transportb_disp_cat_of_strict_twosided_disp_cat_mor _ _ _ (id_left (C := cat_of_setcategory) (functor_identity C))). } cbn -[idtoiso_twosided_disp]. rewrite (two_disp_post_whisker_f (D := D₁)). rewrite (two_disp_pre_whisker_f (D := D₁)). rewrite transport_f_f_disp_mor2. rewrite !(assoc_two_disp_alt (D := D₁)). rewrite !two_disp_post_whisker_f. rewrite !transport_f_f_disp_mor2. etrans. { do 4 apply maponpaths. apply (idtoiso_twosided_disp_concat (D := D₁)). } unfold transportb_disp_mor2. rewrite !two_disp_post_whisker_f. rewrite transport_f_f_disp_mor2. rewrite !(assoc_two_disp (D := D₁)). unfold transportb_disp_mor2. rewrite !transport_f_f_disp_mor2. etrans. { apply maponpaths. do 2 apply maponpaths_2. apply (idtoiso_twosided_disp_concat (D := D₁)). } unfold transportb_disp_mor2. rewrite !two_disp_pre_whisker_f. rewrite transport_f_f_disp_mor2. use transportf_disp_mor2_eq. use idtoiso_twosided_disp_triple_concat. Qed.
imports: UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Core.Setcategories UniMath.CategoryTheory.Categories.CategoryOfSetCategories UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.DisplayedCatEq UniMath.CategoryTheory.DisplayedCats.Examples.DispFunctorPair UniMath.CategoryTheory.TwoSidedDisplayedCats.TwoSidedDispCat UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedFunctor UniMath.CategoryTheory.TwoSidedDisplayedCats.DisplayedNatTrans UniMath.CategoryTheory.TwoSidedDisplayedCats.Isos UniMath.CategoryTheory.TwoSidedDisplayedCats.Univalence UniMath.CategoryTheory.TwoSidedDisplayedCats.Strictness UniMath.CategoryTheory.TwoSidedDisplayedCats.TransportLaws