proposition: Lemma full_disp_naturality_condition {A B C : category} {F : bifunctor A B C} {G : bifunctor A B C} {DA : disp_cat A} {DB : disp_cat B} {DC : disp_cat C} {α : binat_trans F G} {DF : disp_bifunctor_data F DA DB DC} {DG : disp_bifunctor_data G DA DB DC} {dα : disp_binat_trans_data α DF DG} (dαn : is_disp_binat_trans dα) (x1 x2 : A) (y1 y2 : B) (f : A⟦x1,x2⟧) (g : B⟦y1,y2⟧) (xx1 : DA x1) (xx2 : DA x2) (yy1 : DB y1) (yy2 : DB y2) (ff : xx1 -->[f] xx2) (gg : yy1 -->[g] yy2) : (ff ⊗⊗^{DF} gg) ;; (dα _ _ xx2 yy2) = transportb _ (full_naturality_condition (pr2 α) f g) ((dα _ _ xx1 yy1) ;; (ff ⊗⊗^{DG} gg)).
proof: Proof. unfold dispfunctoronmorphisms1. etrans. { apply assoc_disp_var. } apply transportf_transpose_left. rewrite transport_b_b. rewrite (pr1 dαn). etrans. { apply mor_disp_transportf_prewhisker. } apply transportb_transpose_right. rewrite transport_f_f. apply transportf_transpose_left. etrans. { apply assoc_disp. } rewrite (pr2 dαn). apply transportb_transpose_right. rewrite transport_f_b. apply transportf_transpose_left. etrans. { apply mor_disp_transportf_postwhisker. } apply transportf_transpose_left. rewrite transport_b_b. rewrite assoc_disp_var. apply transportf_transpose_left. rewrite transport_b_b. use pathsinv0. apply transportf_set. apply homset_property. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Core.Isos