proposition: 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: Proof. apply pathsinv0. use path_to_ctr. use pathsinv0. rewrite <-BinProductOfArrows_idxcomp. rewrite !assoc'. use cancel_precomposition. use pathsinv0. use (PowerObject_transpose_tri). Defined.
imports: UniMath.Foundations.PartA UniMath.Foundations.PartD UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Limits.Terminal UniMath.CategoryTheory.Limits.BinProducts UniMath.CategoryTheory.SubobjectClassifier UniMath.CategoryTheory.opp_precat UniMath.CategoryTheory.whiskering UniMath.CategoryTheory.PrecategoryBinProduct UniMath.CategoryTheory.OppositeCategory.Core UniMath.CategoryTheory.Categories.HSET.Core UniMath.CategoryTheory.exponentials UniMath.CategoryTheory.Adjunctions.Core