proposition: Lemma comonads_category_Hstandard {F : functor C C} (T : disp_Comonad_data F) (Tlaws : disp_Comonad_laws T) (T' : disp_Comonad_data F) (T'laws : disp_Comonad_laws T') : disp_Comonad_Mor_laws T T' (nat_trans_id F) → disp_Comonad_Mor_laws T' T (nat_trans_id F) → T,, Tlaws = T',, T'laws.
proof: Proof. intros H H'. apply subtypeInjectivity. { intro T0. apply isaprop_disp_Comonad_laws. } cbn. induction T as [δ ε]. induction T' as [δ' ε']. apply dirprodeq; cbn. - apply nat_trans_eq; [apply C|]. intro a. assert (Hinst := pr1 H a). cbn in Hinst. rewrite id_right in Hinst. rewrite id_left in Hinst. rewrite functor_id in Hinst. rewrite id_right in Hinst. exact (!Hinst). - apply nat_trans_eq; [apply C|]. intro a. assert (Hinst := pr2 H a). cbn in Hinst. rewrite id_left in Hinst. exact (!Hinst). Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Notations UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Total UniMath.CategoryTheory.DisplayedCats.Constructions UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.SIP UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.Core.Univalence