proposition: Lemma is_functor_slice_to_fam : is_functor slice_to_fam_data.
proof: Proof. split; [ intro a | intros a b c f g]; apply (nat_trans_eq has_homsets_HSET); intro x; apply funextsec; intro p; apply (invmaponpathsincl pr1); simpl; try (apply isofhlevelfpr1; intro; apply setproperty); repeat (unfold hfiber; rewrite transportf_total2; simpl); repeat (rewrite transportf_const); reflexivity. Qed.
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