Proposition PowerObject_transpose_precomp {a' a b: C} (g : C ⟦a', a⟧)(f : C ⟦(b ⨉ a), Ω⟧ ) : PowerObject_transpose (identity b ⨱ g · f) = g · (PowerObject_transpose f). Proof. apply pathsinv0. use path_to_ctr. use pathsinv0. rewrite <-BinProductOfArrows_idxcomp. rewrite !assoc'. use cancel_precomposition. use pathsinv0. use (PowerObject_transpose_tri). Defined.
Require Import UniMath.Foundations.PartA. Require Import UniMath.Foundations.PartD. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Limits.Terminal. Require Import UniMath.CategoryTheory.Limits.BinProducts. Require Import UniMath.CategoryTheory.SubobjectClassifier.SubobjectClassifier. Require Import UniMath.CategoryTheory.opp_precat. Require Import UniMath.CategoryTheory.whiskering. Require Import UniMath.CategoryTheory.PrecategoryBinProduct. Require Import UniMath.CategoryTheory.OppositeCategory.Core. Require Import UniMath.CategoryTheory.Categories.HSET.Core. Require Import UniMath.CategoryTheory.exponentials. Require Import UniMath.CategoryTheory.Adjunctions.Core.