proposition: Lemma const_prod_functor1_slicecat c (Af : C / c) : constprod_functor1 (BPC c) Af = functor_composite (base_change_functor (pr2 Af)) (slicecat_functor (pr2 Af)).
proof: Proof. apply functor_eq; [apply has_homsets_slice_precat |]. use functor_data_eq. - intro x; apply idpath. - intros x y f; apply (eq_mor_slicecat C); simpl. apply PullbackArrowUnique. + rewrite PullbackArrow_PullbackPr1, id_right; apply idpath. + rewrite PullbackArrow_PullbackPr2; apply idpath. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Univalence UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Limits.Graphs.Limits UniMath.CategoryTheory.Limits.Graphs.Colimits UniMath.CategoryTheory.Limits.Pullbacks UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.Limits.BinCoproducts UniMath.CategoryTheory.Limits.Coproducts UniMath.CategoryTheory.Limits.Initial UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.Monics UniMath.CategoryTheory.Epis