proposition: Lemma disp_mor_unique_disc_fib C (D : discrete_fibration C) : ∏ (c c' : C) (f : c --> c') (d : D c) (d' : D c') (ff ff' : d -->[f] d'), ff = ff'.
proof: Proof. intros. assert (XR := unique_lift f d'). assert (foo : ((d,,ff) : ∑ d0, d0 -->[f] d') = (d,,ff')). { apply proofirrelevancecontr. apply XR. } apply (pair_inj (isaset_fiber_discrete_fibration _ _ ) foo). Defined.
imports: UniMath.Foundations.Sets UniMath.MoreFoundations.All UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.MonoEpiIso UniMath.CategoryTheory.Categories.HSET.Univalence UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.Presheaf UniMath.CategoryTheory.DisplayedCats.Core UniMath.CategoryTheory.DisplayedCats.Functors UniMath.CategoryTheory.DisplayedCats.NaturalTransformations UniMath.CategoryTheory.DisplayedCats.Fiber UniMath.CategoryTheory.DisplayedCats.Isos UniMath.CategoryTheory.DisplayedCats.Univalence UniMath.CategoryTheory.DisplayedCats.Examples.Opposite