Lemma is_functor_LimFunctor_data : is_functor LimFunctor_data. Proof. split. - intro a; simpl. apply pathsinv0, lim_endo_is_identity; intro u. unfold LimFunctor_mor; rewrite limOfArrowsOut. assert (H : # (pr1 (D u)) (identity a) = identity (pr1 (D u) a)). apply (functor_id (D u) a). now rewrite H, id_right. - intros a b c fab fbc; simpl; unfold LimFunctor_mor. apply pathsinv0. eapply pathscomp0; [now apply postcompWithLimOfArrows|]. apply pathsinv0, limArrowUnique; intro u. rewrite limOfArrowsOut, (functor_comp (D u)); simpl. now rewrite <- assoc. Qed.
Require Import UniMath.Foundations.PartD. Require Import UniMath.Foundations.Propositions. Require Import UniMath.Foundations.Sets. Require Import UniMath.MoreFoundations.Tactics. Require Import UniMath.MoreFoundations.Propositions. Require Import UniMath.CategoryTheory.Core.Categories. Require Import UniMath.CategoryTheory.Core.Isos. Require Import UniMath.CategoryTheory.Core.NaturalTransformations. Require Import UniMath.CategoryTheory.Core.Univalence. Require Import UniMath.CategoryTheory.Core.Functors. Require Import UniMath.CategoryTheory.FunctorCategory.