Lemma Comonad_laws_from_adjunction {C D : category} {L : functor C D} {R : functor D C} (H : are_adjoints L R) : disp_Comonad_laws (Comonad_data_from_adjunction H). Proof. cbn. use make_dirprod. + use make_dirprod. * intro c; cbn. apply triangle_id_left_ad. * intro c; cbn. rewrite <- functor_id. rewrite <- functor_comp. apply maponpaths. apply triangle_id_right_ad. + intro c; cbn. do 2 (rewrite <- functor_comp). apply maponpaths. apply pathsinv0. apply (nat_trans_ax ((unit_from_are_adjoints H))). Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Notations. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.FunctorCategory. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.Adjunctions.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Core. Require Import UniMath.CategoryTheory.DisplayedCats.Total. Require Import UniMath.CategoryTheory.DisplayedCats.Constructions. Require Import UniMath.CategoryTheory.DisplayedCats.Isos. Require Import UniMath.CategoryTheory.DisplayedCats.SIP. Require Import UniMath.CategoryTheory.DisplayedCats.Univalence. Require Import UniMath.CategoryTheory.Core.Univalence.