proposition: Lemma φ_adj_after_φ_adj_inv {A : C} {B : D} (g : A --> G B) : φ_adj (φ_adj_inv g) = g.
proof: Proof. unfold φ_adj. unfold φ_adj_inv. assert (X':=triangle_id_right_ad H). rewrite functor_comp. rewrite assoc. assert (X2 := nat_trans_ax η). simpl in X2. rewrite <- X2; clear X2. rewrite <- assoc. intermediate_path (g · identity _). - apply maponpaths. apply X'. - apply id_right. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Epis UniMath.CategoryTheory.Epis UniMath.CategoryTheory.SplitMonicsAndEpis UniMath.CategoryTheory.Monics UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.whiskering