proposition: Lemma pre_comp_z_iso_is_inverse : is_inverse_in_precat (C := D) (η P c) pre_comp_z_iso_inv.
proof: Proof. split. - refine (_ @ functor_id P _). refine (colimArrowCommutes (lan_colim HD F P (F c)) _ _ ((c ,, tt) ,, identity (F c)) @ _). refine (maponpaths (λ f, # (P : _ ⟶ _) f) (_ : invmap (Fweq c c) _ = _)). apply invmap_eq. exact (!functor_id F _). - apply colim_mor_eq. intro v. refine (assoc _ _ _ @ _). match goal with | [ |- _ · ?a = _ ] => refine (maponpaths (λ x, x · a) (colimArrowCommutes _ _ _ _) @ _) end. use (colimInCommutes _ v ((c ,, tt) ,, identity (F c)) ((make_dirprod _ _) ,, _) @ _). + now induction (pr21 v). + refine (maponpaths (λ x, x · _) _). exact (!homotweqinvweq (Fweq _ _) _). + exact (!id_right _). Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.LeftKanExtension UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.Equivalences.FullyFaithful UniMath.CategoryTheory.Adjunctions.Core