Definition bicat_left_adjoint_equivalence_to_op1_bicat_left_adjoint_equivalence {C : bicat} {X Y : op1_bicat C} (f : X --> Y) : @left_adjoint_equivalence C Y X f → left_adjoint_equivalence f. Proof. intros Hf. use tpair. - use tpair. + exact (left_adjoint_right_adjoint Hf). + split. * exact ((left_equivalence_counit_iso Hf)^-1). * exact ((left_equivalence_unit_iso Hf)^-1). - split ; split. + use inv_cell_eq ; cbn. * apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell. is_iso. * apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell. is_iso. * cbn. rewrite !vassocr. exact (internal_triangle1 Hf). + use inv_cell_eq ; cbn. * apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell. is_iso. * apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell. is_iso. * cbn. rewrite !vassocr. exact (internal_triangle2 Hf). + cbn. apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell. is_iso. + cbn. apply bicat_is_invertible_2cell_to_op1_bicat_is_invertible_2cell. is_iso. Defined.
Require Import UniMath.Foundations.All. Require Import UniMath.MoreFoundations.All. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.Bicategories.Core.Bicat. Import Bicat.Notations. Require Import UniMath.Bicategories.Core.Invertible_2cells. Require Import UniMath.Bicategories.Morphisms.Adjunctions.