Lemma uOF_comp {C:category} {X Y Z:[C^op,HSET]} (r:Representation X) (s:Representation Y) (t:Representation Z) (p:X-->Y) (q:Y-->Z) : t \\ ((q ∘ p) ⟳ r) = (t \\ (q ⟳ s)) ∘ (s \\ (p ⟳ r)). Proof. unshelve refine (transportf (λ k, _ \\ k = _) (nattrans_nattrans_arrow_assoc _ _ _) _). unshelve refine (_ @ !univ_arrow_mor_assoc _ _ _). apply maponpaths. unshelve refine (_ @ nattrans_arrow_mor_assoc _ _ _). apply (maponpaths (λ k, q ⟳ k)). apply pathsinv0. apply universalMapProperty. Qed.
Require Import UniMath.Foundations.Preamble. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import Require Import Require Import UniMath.MoreFoundations.Tactics.