Lemma pullback_to_binprod_inv (P : some_pullback C Z) : binprod_to_pullback (pullback_to_binprod P) = P. Proof. induction P as [A [B [f [g P]]]]. unfold binprod_to_pullback , pullback_to_binprod. do 4 (apply (total2_paths2_f (idpath _)); rewrite idpath_transportf). exact (pullback_to_slice_binprod_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.