Lemma binprod_to_pullback_inv (P : some_binprod (C / Z)) : pullback_to_binprod (binprod_to_pullback P) = P. Proof. induction P as [[A f] [[B g] P]]. unfold pullback_to_binprod , binprod_to_pullback. simpl. repeat (apply (total2_paths2_f (idpath _)); rewrite idpath_transportf). exact (slice_binprod_to_pullback_inv P). Qed.
Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.Foundations.Sets. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.slicecat. Require Import UniMath.CategoryTheory.Equivalences.Core. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.Limits.Pullbacks.