proposition: Lemma dialgebra_nat_trans_is_mon_nat_trans : is_mon_nat_trans (comp_fmonoidal_lax dialgebra_monoidal_pr1 Fm) (comp_fmonoidal_lax dialgebra_monoidal_pr1 Gm) (dialgebra_nat_trans F G).
proof: Proof. split. + intros a a'. unfold fmonoidal_preservestensordata. cbn. unfold fmonoidal_preservestensordata. unfold projection_preserves_tensordata. cbn. unfold dialgebra_nat_trans_data. do 2 rewrite functor_id. do 2 rewrite id_right. unfold dialgebra_disp_tensor_op. etrans. { repeat rewrite assoc. do 2 apply cancel_postcomposition. apply (pr12 (fmonoidal_preservestensorstrongly Fm (pr1 a) (pr1 a'))). } rewrite id_left. apply idpath. + unfold is_mon_nat_trans_unitlaw. unfold fmonoidal_preservesunit. cbn. unfold fmonoidal_preservesunit. unfold projection_preserves_unit. cbn. unfold dialgebra_disp_unit. do 2 rewrite functor_id. do 2 rewrite id_right. etrans. { rewrite assoc. apply cancel_postcomposition. apply (pr12 (fmonoidal_preservesunitstrongly Fm)). } unfold fmonoidal_preservesunit. apply id_left. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Categories.Dialgebras UniMath.CategoryTheory.Monoidal.WhiskeredBifunctors UniMath.CategoryTheory.Monoidal.Categories UniMath.CategoryTheory.Monoidal.Functors UniMath.CategoryTheory.Monoidal.Displayed.WhiskeredDisplayedBifunctors UniMath.CategoryTheory.Monoidal.Displayed.Monoidal UniMath.CategoryTheory.Monoidal.Displayed.TotalMonoidal UniMath.CategoryTheory.Monoidal.Displayed.MonoidalSections UniMath.CategoryTheory.Monoidal.Displayed.Symmetric UniMath.CategoryTheory.Monoidal.Displayed.MonoidalFunctorLifting UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.Monoidal.Structure.Symmetric