proposition: Lemma is_functor_LimFunctor_data : is_functor LimFunctor_data.
proof: Proof. split. - intro a; simpl. apply pathsinv0, lim_endo_is_identity; intro u. unfold LimFunctor_mor; rewrite limOfArrowsOut. assert (H : # (pr1 (dob D u)) (identity a) = identity (pr1 (dob D u) a)). apply (functor_id (dob 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 (dob D u)); simpl. now rewrite <- assoc. Qed.
imports: UniMath.Foundations.PartD UniMath.Foundations.Propositions UniMath.Foundations.Sets UniMath.MoreFoundations.Tactics UniMath.MoreFoundations.Propositions UniMath.CategoryTheory.Core.Categories UniMath.CategoryTheory.Core.Functors UniMath.CategoryTheory.Core.Isos UniMath.CategoryTheory.Core.NaturalTransformations UniMath.CategoryTheory.Core.Univalence UniMath.CategoryTheory.Adjunctions.Core UniMath.CategoryTheory.FunctorCategory UniMath.CategoryTheory.Limits.Graphs.Colimits