proposition: Lemma fam_z_iso (F : fam X) : z_iso ((functor_identity (fam X)) F) ((functor_composite fam_to_slice slice_to_fam) F).
proof: Proof. apply (z_iso_from_nat_z_iso has_homsets_HSET). exists ((pr1 fam_unit) F). intro x. exact (hset_equiv_is_z_iso ((pr1 F) x) (make_hSet (hfiber pr1 x) (isaset_hfiber pr1 x (isaset_total2_hSet X (λ x, (pr1 F) x)) (pr2 X))) (ezweqpr1 (funcomp (pr1 (pr1 F)) pr1) x)). Defined.
imports: UniMath.Foundations.Sets UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.Categories.HSET.MonoEpiIso UniMath.CategoryTheory.Categories.HSET.Univalence UniMath.CategoryTheory.Categories.HSET.Slice UniMath.CategoryTheory.slicecat UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.Equivalences.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Groupoids UniMath.CategoryTheory.Categories.StandardCategories