proposition: Proposition form_adj_comp_cod_pb : form_adjunction comp_functor (cod_pb HC f) comp_functor_unit comp_functor_counit.
proof: Proof. split. - intro g. use eq_mor_cod_fib. rewrite comp_in_cod_fib. cbn. rewrite PullbackArrow_PullbackPr1. apply idpath. - intro g. use eq_mor_cod_fib. rewrite comp_in_cod_fib. use (MorphismsIntoPullbackEqual (isPullback_Pullback (HC y (pr1 g) x (pr2 g) f))). + rewrite !assoc'. etrans. { apply maponpaths. apply (PullbackArrow_PullbackPr1 (HC y (pr1 g) x (pr2 g) f)). } rewrite transportf_cod_disp. cbn. rewrite !assoc. rewrite PullbackArrow_PullbackPr1. apply idpath. + rewrite !assoc'. etrans. { apply maponpaths. apply (PullbackArrow_PullbackPr2 (HC y (pr1 g) x (pr2 g) f)). } cbn. rewrite !assoc. rewrite PullbackArrow_PullbackPr2. rewrite id_left, id_right. apply idpath. Qed.
imports: UniMath.Foundations.All UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Prelude UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Fibrations UniMath.CategoryTheory.DisplayedCats.Codomain UniMath.CategoryTheory.DisplayedCats.Codomain.FiberCod UniMath.CategoryTheory.DisplayedCats.DependentSums