proposition: Lemma slice_fam_form_adj : form_adjunction fam_to_slice slice_to_fam fam_unit slice_counit.
proof: Proof. unfold form_adjunction. split. + intro f. apply eq_mor_slicecat. apply funextsec. intro x. reflexivity. + intro F. apply (nat_trans_eq has_homsets_HSET). intro x. apply funextsec. intro f. apply (invmaponpathsincl pr1). - apply isofhlevelfpr1. intro. apply setproperty. - simpl. unfold hfiber. rewrite transportf_total2. simpl. rewrite transportf_const. reflexivity. 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